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| Boilerpipe Text | In
trigonometry
,
trigonometric identities
are
equalities
that involve
trigonometric functions
and are true for every value of the occurring
variables
for which both sides of the equality are defined. Geometrically, these are
identities
involving certain functions of one or more
angles
. They are distinct from
triangle identities
, which are identities potentially involving angles but also involving side lengths or other lengths of a
triangle
.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the
integration
of non-trigonometric functions: a common technique involves first using the
substitution rule with a trigonometric function
, and then simplifying the resulting integral with a trigonometric identity.
Pythagorean identities
[
edit
]
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity
, and the red triangle shows that
.
The basic relationship between the
sine and cosine
is given by the Pythagorean identity:
where
means
and
means
This can be viewed as a version of the
Pythagorean theorem
, and follows from the equation
for the
unit circle
. This equation can be solved for either the sine or the cosine:
where the sign depends on the
quadrant
of
Dividing this identity by
,
, or both yields the following identities:
Using these identities, it is possible to express any trigonometric function in terms of any other (
up to
a plus or minus sign):
Each trigonometric function in terms of each of the other five.
[
1
]
in terms of
Reflections, shifts, and periodicity
[
edit
]
By examining the unit circle, one can establish the following properties of the trigonometric functions.
Transformation of coordinates (
a
,
b
) when shifting the reflection angle
in increments of
When the direction of a
Euclidean vector
is represented by an angle
this is the angle determined by the free vector (starting at the origin) and the positive
-unit vector. The same concept may also be applied to lines in an
Euclidean space
, where the angle is that determined by a parallel to the given line through the origin and the positive
-axis. If a line (vector) with direction
is reflected about a line with direction
then the direction angle
of this reflected line (vector) has the value
The values of the trigonometric functions of these angles
for specific angles
satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as
reduction formulae
.
[
2
]
reflected in
[
3
]
odd/even
identities
reflected in
reflected in
reflected in
reflected in
compare to
Shifts and periodicity
[
edit
]
Transformation of coordinates (
a
,
b
) when shifting the angle
in increments of
Shift by one quarter period
Shift by one half period
Shift by full periods
[
4
]
Period
The sign of trigonometric functions depends on quadrant of the angle. If
and
sgn
is the
sign function
,
The trigonometric functions are periodic with common period
so for values of
θ
outside the interval
they take repeating values (see
§ Shifts and periodicity
above).
The sign of a sinusoid or cosinusoid can be used to define a normalized
square wave
.
For example, the functions
and
take values
±1
and correspond to square waves with a phase shift of
π
/
2
.
Angle sum and difference identities
[
edit
]
Geometric construction to derive angle sum trigonometric identities
Diagram showing the angle difference identities for
and
These are also known as the
angle addition and subtraction theorems
(or
formulae
).
The angle difference identities for
and
can be derived from the angle sum versions (and vice versa) by substituting
for
and using the facts that
and
They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here. They can also be seen as expressing the
dot product
and
cross product
of two vectors in terms of the cosine and the sine of the angle between them.
These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.
Sine
[
5
]
[
6
]
Cosine
[
6
]
[
7
]
Tangent
[
6
]
[
8
]
Cosecant
[
9
]
Secant
[
9
]
Cotangent
[
6
]
[
10
]
Arcsine
[
11
]
Arccosine
[
12
]
Arctangent
[
13
]
Arccotangent
Sines and cosines of sums of infinitely many angles
[
edit
]
When the series
converges absolutely
then
Because the series
converges absolutely, it is necessarily the case that
and
Particularly, in these two identities, an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are
cofinitely
many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.
When only finitely many of the angles
are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
Tangents and cotangents of sums
[
edit
]
Let
(for
) be the
k
th-degree
elementary symmetric polynomial
in the variables
for
that is,
Then
This can be shown by using the sine and cosine sum formulae above:
The number of terms on the right side depends on the number of terms on the left side.
For example:
and so on. The case of only finitely many terms can be proved by
mathematical induction
.
[
14
]
The case of infinitely many terms can be proved by using some elementary inequalities.
[
15
]
Suppose
and
and
and let
be any number for which
Suppose that
so that the forgoing fraction cannot be
0
/
0
. Then for all
[
16
]
(In case the denominator of this fraction is 0, we take the value of the fraction to be
, where the symbol
does not mean either
or
, but is the
that is approached by going in either the positive or the negative direction, making the completion of the line
topologically a circle.)
From this identity it can be shown to follow quickly that the family of all
Cauchy-distributed
random variables is closed under linear fractional transformations, a result known since 1976.
[
17
]
Secants and cosecants of sums
[
edit
]
where
is the
k
th-degree
elementary symmetric polynomial
in the
n
variables
and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.
[
18
]
The case of only finitely many terms can be proved by mathematical induction on the number of such terms.
For example,
Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine. Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine:
sin(
α
+
β
) = sin
α
cos
β
+ cos
α
sin
β
.
Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a
cyclic quadrilateral
, as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.
[
19
]
The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.
By
Thales's theorem
,
and
are both right angles. The right-angled triangles
and
both share the hypotenuse
of length 1. Thus, the side
,
,
and
.
By the
inscribed angle
theorem, the
central angle
subtended by the chord
at the circle's center is twice the angle
, i.e.
. Therefore, the symmetrical pair of red triangles each has the angle
at the center. Each of these triangles has a
hypotenuse
of length
, so the length of
is
, i.e. simply
. The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also
.
When these values are substituted into the statement of Ptolemy's theorem that
, this yields the angle sum trigonometric identity for sine:
. The angle difference formula for
can be similarly derived by letting the side
serve as a diameter instead of
.
[
19
]
Multiple-angle and half-angle formulas
[
edit
]
T
n
is the
n
th
Chebyshev polynomial
[
20
]
de Moivre's formula
,
i
is the
imaginary unit
[
21
]
Multiple-angle formulae
[
edit
]
Double-angle formulae
[
edit
]
Visual demonstration of the double-angle formula for sine. For the above isosceles triangle with unit sides and angle
, the area
1
/
2
× base × height is calculated in two orientations. When upright, the area is
. When on its side, the same area is
. Therefore,
Formulae for twice an angle.
[
22
]
[
23
]
Triple-angle formulae
[
edit
]
Formulae for triple angles.
[
22
]
Multiple-angle formulae
[
edit
]
Formulae for multiple angles.
[
24
]
The
Chebyshev
method is a
recursive
algorithm
for finding the
n
th multiple angle formula knowing the
th and
th values.
[
25
]
can be computed from
,
, and
with
This can be proved by adding together the formulae
It follows by induction that
is a polynomial of
the so-called Chebyshev polynomial of the first kind, see
Chebyshev polynomials#Trigonometric definition
.
Similarly,
can be computed from
and
with
This can be proved by adding formulae for
and
Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:
Half-angle formulas
[
edit
]
[
26
]
[
27
]
Also
These can be shown by using either the sum and difference identities or the multiple-angle formulae.
Sine
Cosine
Tangent
Cotangent
Double-angle formula
[
28
]
[
29
]
Triple-angle formula
[
20
]
[
30
]
Half-angle formula
[
26
]
[
27
]
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a
compass and straightedge construction
of
angle trisection
to the algebraic problem of solving a
cubic equation
, which allows one to prove that
trisection is in general impossible
using the given tools.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the
cubic equation
4
x
3
− 3
x
+
d
= 0
, where
x
is the value of the cosine function at the one-third angle and
d
is the known value of the cosine function at the full angle. However, the
discriminant
of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle).
None of these solutions are reducible
to a real
algebraic expression
, as they use intermediate complex numbers under the
cube roots
.
Power-reduction formulae
[
edit
]
Obtained by solving the second and third versions of the cosine double-angle formula.
Sine
Cosine
Other
In general terms of powers of
or
the following is true, and can be deduced using
De Moivre's formula
,
Euler's formula
and the
binomial theorem
.
if
n
is ...
n
is odd
n
is even
Product-to-sum and sum-to-product identities
[
edit
]
Proof of the sum-and-difference-to-product cosine identity for prosthaphaeresis calculations using an
isosceles triangle
The product-to-sum identities
[
31
]
or
prosthaphaeresis
formulae can be proven by expanding their right-hand sides using the
angle addition theorems
. Historically, the first four of these were known as
Werner's formulas
, after
Johannes Werner
who used them for astronomical calculations.
[
32
]
See
amplitude modulation
for an application of the product-to-sum formulae, and
beat (acoustics)
and
phase detector
for applications of the sum-to-product formulae.
Product-to-sum identities
[
edit
]
The product of two sines or cosines of different angles can be converted to a sum of trigonometric functions of a sum and difference of those angles:
As a corollary, the product or quotient of tangents can be converted to a quotient of sums of cosines or sines, respectively,
More generally, for a product of any number of sines or cosines,
[
citation needed
]
Sum-to-product identities
[
edit
]
Diagram illustrating sum-to-product identities for sine and cosine. The blue right-angled triangle has angle
and the red right-angled triangle has angle
. Both have a hypotenuse of length 1. Auxiliary angles, here called
and
, are constructed such that
and
. Therefore,
and
. This allows the two congruent purple-outline triangles
and
to be constructed, each with hypotenuse
and angle
at their base. The sum of the heights of the red and blue triangles is
, and this is equal to twice the height of one purple triangle, i.e.
. Writing
and
in that equation in terms of
and
yields a sum-to-product identity for sine:
. Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.
The sum of sines or cosines of two angles can be converted to a product of sines or cosines of the mean and half the difference of the angles:
[
33
]
The sum of the tangent of two angles can be converted to a quotient of the sine of angles divided by the product of the cosines:
[
33
]
Hermite's cotangent identity
[
edit
]
Charles Hermite
demonstrated the following identity.
[
34
]
Suppose
are
complex numbers
, no two of which differ by an integer multiple of
π
. Let
(in particular,
being an
empty product
, is 1). Then
The simplest non-trivial example is the case
n
= 2
:
Finite products of trigonometric functions
[
edit
]
For
coprime
integers
n
,
m
where
T
n
is the
Chebyshev polynomial
.
[
citation needed
]
The following relationship holds for the sine function
More generally for an integer
n
> 0
[
35
]
or written in terms of the
chord
function
,
This comes from the
factorization of the polynomial
into linear factors (cf.
root of unity
): For any complex
z
and an integer
n
> 0
,
Linear combinations
[
edit
]
For some purposes it is important to know that any
linear combination
of sine waves of the same period or frequency but different
phase shifts
is also a sine wave with the same period or frequency, but a different phase shift. This is useful in
sinusoid
data fitting
, because the measured or observed data are linearly related to the
a
and
b
unknowns of the
in-phase and quadrature components
basis below, resulting in a simpler
Jacobian
, compared to that of
and
.
The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,
[
36
]
[
37
]
where
and
are defined as so:
given that
Arbitrary phase shift
[
edit
]
More generally, for arbitrary phase shifts, we have
where
and
satisfy:
More than two sinusoids
[
edit
]
The general case reads
[
37
]
where
and
Lagrange's trigonometric identities
[
edit
]
These identities, named after
Joseph Louis Lagrange
, are:
[
38
]
[
39
]
[
40
]
for
A related function is the
Dirichlet kernel
:
A similar identity is
[
41
]
The proof is the following. By using the
angle sum and difference identities
,
Then let's examine the following formula,
and this formula can be written by using the above identity,
So, dividing this formula with
completes the proof.
Certain linear fractional transformations
[
edit
]
If
is given by the
linear fractional transformation
and similarly
then
More tersely stated, if for all
we let
be what we called
above, then
If
is the slope of a line, then
is the slope of its rotation through an angle of
Relation to the complex exponential function
[
edit
]
Euler's formula states that, for any real number
x
:
[
42
]
where
i
is the
imaginary unit
. Substituting −
x
for
x
gives us:
These two equations can be used to solve for cosine and sine in terms of the
exponential function
. Specifically,
[
43
]
[
44
]
These formulae are useful for proving many other trigonometric identities. For example, that
e
i
(
θ
+
φ
)
=
e
iθ
e
iφ
means that
cos(
θ
+
φ
) +
i
sin(
θ
+
φ
) = (cos
θ
+
i
sin
θ
) (cos
φ
+
i
sin
φ
) = (cos
θ
cos
φ
− sin
θ
sin
φ
) +
i
(cos
θ
sin
φ
+ sin
θ
cos
φ
)
.
That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.
The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the
complex logarithm
.
Function
Inverse function
[
45
]
Relation to complex hyperbolic functions
[
edit
]
Trigonometric functions may be deduced from
hyperbolic functions
with
complex
arguments. The formulae for the relations are shown below
[
46
]
[
47
]
.
When using a
power series
expansion to define trigonometric functions, the following identities are obtained:
[
48
]
Infinite product formulae
[
edit
]
For applications to
special functions
, the following
infinite product
formulae for trigonometric functions are useful:
[
49
]
[
50
]
Inverse trigonometric functions
[
edit
]
The following identities give the result of composing a trigonometric function with an inverse trigonometric function.
[
51
]
Taking the
multiplicative inverse
of both sides of the each equation above results in the equations for
The right hand side of the formula above will always be flipped.
For example, the equation for
is:
while the equations for
and
are:
The following identities are implied by the
reflection identities
. They hold whenever
and
are in the domains of the relevant functions.
Also,
[
52
]
The
arctangent
function can be expanded as a series:
[
53
]
Identities without variables
[
edit
]
In terms of the
arctangent
function we have
[
52
]
The curious identity known as
Morrie's law
,
is a special case of an identity that contains one variable:
Similarly,
is a special case of an identity with
:
For the case
,
For the case
,
The same cosine identity is
Similarly,
Similarly,
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than
21
/
2
that are
relatively prime
to (or have no
prime factors
in common with) 21. The last several examples are corollaries of a basic fact about the irreducible
cyclotomic polynomials
: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the
Möbius function
evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
Other cosine identities include:
[
54
]
and so forth for all odd numbers, and hence
Many of those curious identities stem from more general facts like the following:
[
55
]
and
Combining these gives us
If
n
is an odd number (
) we can make use of the symmetries to get
The transfer function of the
Butterworth low pass filter
can be expressed in terms of polynomial and poles. By setting the frequency as the
cutoff frequency
, the following identity can be proved:
An efficient way to
compute
π
to a
large number of digits
is based on the following identity without variables, due to
Machin
. This is known as a
Machin-like formula
:
or, alternatively, by using an identity of
Leonhard Euler
:
or by using
Pythagorean triples
:
Others include:
[
56
]
[
52
]
Generally, for numbers
t
1
, ...,
t
n
−1
∈ (−1, 1)
for which
θ
n
= Σ
n
−1
k
=1
arctan
t
k
∈ (
π
/4, 3
π
/4)
, let
t
n
= tan(
π
/2 −
θ
n
) = cot
θ
n
. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are
t
1
, ...,
t
n
−1
and its value will be in
(−1, 1)
. In particular, the computed
t
n
will be rational whenever all the
t
1
, ...,
t
n
−1
values are rational. With these values,
where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the
t
k
values is not within
(−1, 1)
. Note that if
t
=
p
/
q
is rational, then the
(2
t
, 1 −
t
2
, 1 +
t
2
)
values in the above formulae are proportional to the Pythagorean triple
(2
pq
,
q
2
−
p
2
,
q
2
+
p
2
)
.
For example, for
n
= 3
terms,
for any
a
,
b
,
c
,
d
> 0
.
An identity of Euclid
[
edit
]
Euclid
showed in Book XIII, Proposition 10 of his
Elements
that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:
Ptolemy
used this proposition to compute some angles in
his table of chords
in Book I, chapter 11 of
Almagest
.
Composition of trigonometric functions
[
edit
]
These identities involve a trigonometric function of a trigonometric function:
[
57
]
where
J
i
are
Bessel functions
.
Further "conditional" identities for the case
α
+
β
+
γ
= 180°
[
edit
]
A
conditional trigonometric identity
is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied.
[
58
]
The following formulae apply to arbitrary plane triangles and follow from
as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).
[
59
]
Historical shorthands
[
edit
]
The
versine
,
coversine
,
haversine
, and
exsecant
were used in navigation. For example, the
haversine formula
was used to calculate the distance between two points on a sphere. They are rarely used today.
The
Dirichlet kernel
D
n
(
x
)
is the function occurring on both sides of the next identity:
The
convolution
of any
integrable function
of period
with the Dirichlet kernel coincides with the function's
th-degree Fourier approximation. The same holds for any
measure
or
generalized function
.
Tangent half-angle substitution
[
edit
]
If we set
then
[
60
]
where
sometimes abbreviated to
cis
x
.
When this substitution of
for
tan
x
/
2
is used in
calculus
, it follows that
is replaced by
2
t
/
1 +
t
2
,
is replaced by
1 −
t
2
/
1 +
t
2
and the differential
d
x
is replaced by
2 d
t
/
1 +
t
2
. Thereby one converts rational functions of
and
to rational functions of
in order to find their
antiderivatives
.
Viète's infinite product
[
edit
]
Aristarchus's inequality
Derivatives of trigonometric functions
Exact trigonometric values
(values of sine and cosine expressed in surds)
Exsecant
Half-side formula
Hyperbolic function
Laws for solution of triangles:
Law of cosines
Spherical law of cosines
Law of sines
Law of tangents
Law of cotangents
Mollweide's formula
List of integrals of trigonometric functions
Mnemonics in trigonometry
Pentagramma mirificum
Proofs of trigonometric identities
Prosthaphaeresis
Pythagorean theorem
Tangent half-angle formula
Trigonometric number
Trigonometry
Uses of trigonometry
Versine
and
haversine
^
Abramowitz, Milton
;
Stegun, Irene Ann
, eds. (1983) [June 1964].
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Values of sin and cos, expressed in surds, for integer multiples of 3° and of
5
+
5
/
8
°
, and for the same angles
csc and sec
and
tan |
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## Contents
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- [(Top)](https://en.wikipedia.org/wiki/List_of_trigonometric_identities)
- [1 Pythagorean identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Pythagorean_identities)
- [2 Reflections, shifts, and periodicity](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Reflections,_shifts,_and_periodicity)
Toggle Reflections, shifts, and periodicity subsection
- [2\.1 Reflections](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Reflections)
- [2\.2 Shifts and periodicity](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Shifts_and_periodicity)
- [2\.3 Signs](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Signs)
- [3 Angle sum and difference identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities)
Toggle Angle sum and difference identities subsection
- [3\.1 Sines and cosines of sums of infinitely many angles](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sines_and_cosines_of_sums_of_infinitely_many_angles)
- [3\.2 Tangents and cotangents of sums](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangents_and_cotangents_of_sums)
- [3\.3 Linear fractional transformations of tangents, related to tangents of sums](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_fractional_transformations_of_tangents,_related_to_tangents_of_sums)
- [3\.4 Secants and cosecants of sums](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Secants_and_cosecants_of_sums)
- [3\.5 Ptolemy's theorem](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Ptolemy's_theorem)
- [4 Multiple-angle and half-angle formulas](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_and_half-angle_formulas)
Toggle Multiple-angle and half-angle formulas subsection
- [4\.1 Multiple-angle formulae](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_formulae)
- [4\.1.1 Double-angle formulae](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle_formulae)
- [4\.1.2 Triple-angle formulae](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Triple-angle_formulae)
- [4\.1.3 Multiple-angle formulae](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_formulae_2)
- [4\.1.4 Chebyshev method](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Chebyshev_method)
- [4\.2 Half-angle formulas](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Half-angle_formulas)
- [4\.3 Table](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Table)
- [5 Power-reduction formulae](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulae)
- [6 Product-to-sum and sum-to-product identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities)
Toggle Product-to-sum and sum-to-product identities subsection
- [6\.1 Product-to-sum identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_identities)
- [6\.2 Sum-to-product identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sum-to-product_identities)
- [6\.3 Hermite's cotangent identity](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Hermite's_cotangent_identity)
- [6\.4 Finite products of trigonometric functions](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Finite_products_of_trigonometric_functions)
- [7 Linear combinations](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinations)
Toggle Linear combinations subsection
- [7\.1 Sine and cosine](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sine_and_cosine)
- [7\.2 Arbitrary phase shift](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Arbitrary_phase_shift)
- [7\.3 More than two sinusoids](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#More_than_two_sinusoids)
- [8 Lagrange's trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Lagrange's_trigonometric_identities)
- [9 Certain linear fractional transformations](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Certain_linear_fractional_transformations)
- [10 Relation to the complex exponential function](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Relation_to_the_complex_exponential_function)
- [11 Relation to complex hyperbolic functions](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Relation_to_complex_hyperbolic_functions)
- [12 Series expansion](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Series_expansion)
- [13 Infinite product formulae](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Infinite_product_formulae)
- [14 Inverse trigonometric functions](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Inverse_trigonometric_functions)
- [15 Identities without variables](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Identities_without_variables)
Toggle Identities without variables subsection
- [15\.1 Computing π](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Computing_%CF%80)
- [15\.2 An identity of Euclid](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#An_identity_of_Euclid)
- [16 Composition of trigonometric functions](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Composition_of_trigonometric_functions)
- [17 Further "conditional" identities for the case *α* + *β* + *γ* = 180°](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Further_"conditional"_identities_for_the_case_%CE%B1_+_%CE%B2_+_%CE%B3_=_180%C2%B0)
- [18 Historical shorthands](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Historical_shorthands)
- [19 Miscellaneous](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Miscellaneous)
Toggle Miscellaneous subsection
- [19\.1 Dirichlet kernel](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Dirichlet_kernel)
- [19\.2 Tangent half-angle substitution](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangent_half-angle_substitution)
- [19\.3 Viète's infinite product](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Vi%C3%A8te's_infinite_product)
- [20 See also](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#See_also)
- [21 References](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#References)
- [22 Bibliography](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Bibliography)
- [23 External links](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#External_links)
Toggle the table of contents
# List of trigonometric identities
43 languages
- [العربية](https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D8%A6%D9%85%D8%A9_%D8%A7%D9%84%D9%85%D8%AA%D8%B7%D8%A7%D8%A8%D9%82%D8%A7%D8%AA_%D8%A7%D9%84%D9%85%D8%AB%D9%84%D8%AB%D9%8A%D8%A9 "قائمة المتطابقات المثلثية – Arabic")
- [Azərbaycanca](https://az.wikipedia.org/wiki/Triqonometriyan%C4%B1n_%C9%99sas_d%C3%BCsturlar%C4%B1 "Triqonometriyanın əsas düsturları – Azerbaijani")
- [Беларуская](https://be.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%B3%D0%B0%D0%BD%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%87%D0%BD%D1%8B%D1%8F_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D1%8B "Трыганаметрычныя формулы – Belarusian")
- [Български](https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8_%D1%82%D1%8A%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%B0 "Тригонометрични тъждества – Bulgarian")
- [বাংলা](https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A7%8B%E0%A6%A3%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%85%E0%A6%AD%E0%A7%87%E0%A6%A6%E0%A6%B8%E0%A6%AE%E0%A7%82%E0%A6%B9%E0%A7%87%E0%A6%B0_%E0%A6%A4%E0%A6%BE%E0%A6%B2%E0%A6%BF%E0%A6%95%E0%A6%BE "ত্রিকোণমিতিক অভেদসমূহের তালিকা – Bangla")
- [Català](https://ca.wikipedia.org/wiki/Llista_d%27identitats_trigonom%C3%A8triques "Llista d'identitats trigonomètriques – Catalan")
- [کوردی](https://ckb.wikipedia.org/wiki/%D9%BE%DB%8E%DA%95%D8%B3%D8%AA%DB%8C_%DA%BE%D8%A7%D9%88%D8%A6%DB%95%D9%86%D8%AC%D8%A7%D9%85%DB%95_%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%DB%8C%DB%8C%DB%8C%DB%95%DA%A9%D8%A7%D9%86 "پێڕستی ھاوئەنجامە سێگۆشەیییەکان – Central Kurdish")
- [Чӑвашла](https://cv.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%80%D0%B8_%C3%A7%D0%B0%D0%B2%D0%B0%D1%85%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BC "Тригонометрири çавахлăхсем – Chuvash")
- [Cymraeg](https://cy.wikipedia.org/wiki/Rhestr_unfathiannau_trigonometrig "Rhestr unfathiannau trigonometrig – Welsh")
- [Deutsch](https://de.wikipedia.org/wiki/Formelsammlung_Trigonometrie "Formelsammlung Trigonometrie – German")
- [Español](https://es.wikipedia.org/wiki/Anexo:Identidades_trigonom%C3%A9tricas "Anexo:Identidades trigonométricas – Spanish")
- [فارسی](https://fa.wikipedia.org/wiki/%D9%81%D9%87%D8%B1%D8%B3%D8%AA_%D8%A7%D8%AA%D8%AD%D8%A7%D8%AF%D9%87%D8%A7%DB%8C_%D9%85%D8%AB%D9%84%D8%AB%D8%A7%D8%AA%DB%8C "فهرست اتحادهای مثلثاتی – Persian")
- [Français](https://fr.wikipedia.org/wiki/Formule_de_trigonom%C3%A9trie "Formule de trigonométrie – French")
- [Galego](https://gl.wikipedia.org/wiki/Lista_de_identidades_trigonom%C3%A9tricas "Lista de identidades trigonométricas – Galician")
- [עברית](https://he.wikipedia.org/wiki/%D7%96%D7%94%D7%95%D7%99%D7%95%D7%AA_%D7%98%D7%A8%D7%99%D7%92%D7%95%D7%A0%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%95%D7%AA "זהויות טריגונומטריות – Hebrew")
- [हिन्दी](https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%B8%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A4%BE%E0%A4%93%E0%A4%82_%E0%A4%95%E0%A5%80_%E0%A4%B8%E0%A5%82%E0%A4%9A%E0%A5%80 "त्रिकोणमितीय सर्वसमिकाओं की सूची – Hindi")
- [Hrvatski](https://hr.wikipedia.org/wiki/Popis_trigonometrijskih_jednakosti "Popis trigonometrijskih jednakosti – Croatian")
- [Magyar](https://hu.wikipedia.org/wiki/Trigonometrikus_azonoss%C3%A1gok "Trigonometrikus azonosságok – Hungarian")
- [Հայերեն](https://hy.wikipedia.org/wiki/%D4%B5%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6%D5%A1%D5%B9%D5%A1%D6%83%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B6%D5%B8%D6%82%D5%B5%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80 "Եռանկյունաչափական նույնություններ – Armenian")
- [Bahasa Indonesia](https://id.wikipedia.org/wiki/Daftar_identitas_trigonometri "Daftar identitas trigonometri – Indonesian")
- [Italiano](https://it.wikipedia.org/wiki/Identit%C3%A0_trigonometrica "Identità trigonometrica – Italian")
- [日本語](https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0%E3%81%AE%E5%85%AC%E5%BC%8F%E3%81%AE%E4%B8%80%E8%A6%A7 "三角関数の公式の一覧 – Japanese")
- [Қазақша](https://kk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D2%AF%D0%B9%D0%BB%D0%B5%D1%81%D1%96%D0%BC%D0%B4%D1%96%D0%BA%D1%82%D0%B5%D1%80 "Тригонометриялық үйлесімдіктер – Kazakh")
- [한국어](https://ko.wikipedia.org/wiki/%EC%82%BC%EA%B0%81_%ED%95%A8%EC%88%98_%ED%95%AD%EB%93%B1%EC%8B%9D "삼각 함수 항등식 – Korean")
- [Македонски](https://mk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D1%81%D0%BE%D0%BA_%D0%BD%D0%B0_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%81%D0%BA%D0%B8_%D0%B5%D0%B4%D0%BD%D0%B0%D0%BA%D0%B2%D0%BE%D1%81%D1%82%D0%B8 "Список на тригонометриски еднаквости – Macedonian")
- [Nederlands](https://nl.wikipedia.org/wiki/Lijst_van_goniometrische_gelijkheden "Lijst van goniometrische gelijkheden – Dutch")
- [Norsk bokmål](https://no.wikipedia.org/wiki/Liste_over_trigonometriske_identiteter "Liste over trigonometriske identiteter – Norwegian Bokmål")
- [Polski](https://pl.wikipedia.org/wiki/To%C5%BCsamo%C5%9Bci_trygonometryczne "Tożsamości trygonometryczne – Polish")
- [Português](https://pt.wikipedia.org/wiki/Identidade_trigonom%C3%A9trica "Identidade trigonométrica – Portuguese")
- [Română](https://ro.wikipedia.org/wiki/Identit%C4%83%C8%9Bi_trigonometrice "Identități trigonometrice – Romanian")
- [Руски](https://rsk.wikipedia.org/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D0%B8_%D1%82%D1%80%D0%B8%D2%91%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%B9%D0%BD%D0%B8_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B8 "Основни триґонометрийни формули – Pannonian Rusyn")
- [Русский](https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%82%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%B0 "Тригонометрические тождества – Russian")
- [Саха тыла](https://sah.wikipedia.org/wiki/%D0%A1%D2%AF%D1%80%D2%AF%D0%BD_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D0%B9_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0%D0%BB%D0%B0%D1%80 "Сүрүн тригонометрическай формулалар – Yakut")
- [Српски / srpski](https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D1%81%D0%BA%D0%B8_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%82%D0%B5%D1%82%D0%B8 "Тригонометријски идентитети – Serbian")
- [Svenska](https://sv.wikipedia.org/wiki/Lista_%C3%B6ver_trigonometriska_identiteter "Lista över trigonometriska identiteter – Swedish")
- [தமிழ்](https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%8A%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AF%88%E0%AE%95%E0%AE%B3%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%AA%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D "முக்கோணவியல் முற்றொருமைகளின் பட்டியல் – Tamil")
- [ไทย](https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B2%E0%B8%A2%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%AD%E0%B8%81%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B9%8C%E0%B8%95%E0%B8%A3%E0%B8%B5%E0%B9%82%E0%B8%81%E0%B8%93%E0%B8%A1%E0%B8%B4%E0%B8%95%E0%B8%B4 "รายการเอกลักษณ์ตรีโกณมิติ – Thai")
- [Türkçe](https://tr.wikipedia.org/wiki/Trigonometrik_%C3%B6zde%C5%9Flikler_listesi "Trigonometrik özdeşlikler listesi – Turkish")
- [Українська](https://uk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D1%81%D0%BE%D0%BA_%D1%82%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8%D1%85_%D1%82%D0%BE%D1%82%D0%BE%D0%B6%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9 "Список тригонометричних тотожностей – Ukrainian")
- [Oʻzbekcha / ўзбекча](https://uz.wikipedia.org/wiki/Trigonometrik_ayniyatlar "Trigonometrik ayniyatlar – Uzbek")
- [Tiếng Việt](https://vi.wikipedia.org/wiki/%C4%90%E1%BA%B3ng_th%E1%BB%A9c_l%C6%B0%E1%BB%A3ng_gi%C3%A1c "Đẳng thức lượng giác – Vietnamese")
- [粵語](https://zh-yue.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8%E5%85%AC%E5%BC%8F%E4%B8%80%E8%A6%BD "三角函數公式一覽 – Cantonese")
- [中文](https://zh.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E6%81%92%E7%AD%89%E5%BC%8F "三角恒等式 – Chinese")
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From Wikipedia, the free encyclopedia
| [Trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry") |
|---|
| [](https://en.wikipedia.org/wiki/File:Trig_Functions.svg) |
| [Outline](https://en.wikipedia.org/wiki/Outline_of_trigonometry "Outline of trigonometry") [History](https://en.wikipedia.org/wiki/History_of_trigonometry "History of trigonometry") [Usage](https://en.wikipedia.org/wiki/Uses_of_trigonometry "Uses of trigonometry") [Functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions") ([sin](https://en.wikipedia.org/wiki/Sine_and_cosine "Sine and cosine"), [cos](https://en.wikipedia.org/wiki/Sine_and_cosine "Sine and cosine"), [tan](https://en.wikipedia.org/wiki/Trigonometric_functions#tangent "Trigonometric functions"), [inverse](https://en.wikipedia.org/wiki/Inverse_trigonometric_functions "Inverse trigonometric functions")) [Generalized trigonometry](https://en.wikipedia.org/wiki/Generalized_trigonometry "Generalized trigonometry") |
| Reference |
| [Identities]() [Exact constants](https://en.wikipedia.org/wiki/Exact_trigonometric_values "Exact trigonometric values") [Tables](https://en.wikipedia.org/wiki/Trigonometric_tables "Trigonometric tables") [Unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle") |
| Laws and theorems |
| [Sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines") [Cosines](https://en.wikipedia.org/wiki/Law_of_cosines "Law of cosines") [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem") [Tangents](https://en.wikipedia.org/wiki/Law_of_tangents "Law of tangents") [Cotangents](https://en.wikipedia.org/wiki/Law_of_cotangents "Law of cotangents") |
| [Calculus](https://en.wikipedia.org/wiki/Calculus "Calculus") |
| [Trigonometric substitution](https://en.wikipedia.org/wiki/Trigonometric_substitution "Trigonometric substitution") [Integrals](https://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions "List of integrals of trigonometric functions") ([inverse functions](https://en.wikipedia.org/wiki/List_of_integrals_of_inverse_trigonometric_functions "List of integrals of inverse trigonometric functions")) [Derivatives](https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions "Differentiation of trigonometric functions") [Trigonometric series](https://en.wikipedia.org/wiki/Trigonometric_series "Trigonometric series") |
| Mathematicians |
| [Hipparchus](https://en.wikipedia.org/wiki/Hipparchus "Hipparchus") [Ptolemy](https://en.wikipedia.org/wiki/Ptolemy "Ptolemy") [Brahmagupta](https://en.wikipedia.org/wiki/Brahmagupta "Brahmagupta") [al-Hasib](https://en.wikipedia.org/wiki/Habash_al-Hasib_al-Marwazi "Habash al-Hasib al-Marwazi") [al-Battani](https://en.wikipedia.org/wiki/Al-Battani "Al-Battani") [Regiomontanus](https://en.wikipedia.org/wiki/Regiomontanus "Regiomontanus") [Viète](https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te "François Viète") [de Moivre](https://en.wikipedia.org/wiki/Abraham_de_Moivre "Abraham de Moivre") [Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler") [Fourier](https://en.wikipedia.org/wiki/Joseph_Fourier "Joseph Fourier") |
| [v](https://en.wikipedia.org/wiki/Template:Trigonometry "Template:Trigonometry") [t](https://en.wikipedia.org/wiki/Template_talk:Trigonometry "Template talk:Trigonometry") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Trigonometry "Special:EditPage/Template:Trigonometry") |
In [trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry"), **trigonometric identities** are [equalities](https://en.wikipedia.org/wiki/Equality_\(mathematics\) "Equality (mathematics)") that involve [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions") and are true for every value of the occurring [variables](https://en.wikipedia.org/wiki/Variable_\(mathematics\) "Variable (mathematics)") for which both sides of the equality are defined. Geometrically, these are [identities](https://en.wikipedia.org/wiki/Identity_\(mathematics\) "Identity (mathematics)") involving certain functions of one or more [angles](https://en.wikipedia.org/wiki/Angle "Angle"). They are distinct from [triangle identities](https://en.wikipedia.org/wiki/Trigonometry#Triangle_identities "Trigonometry"), which are identities potentially involving angles but also involving side lengths or other lengths of a [triangle](https://en.wikipedia.org/wiki/Triangle "Triangle").
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the [integration](https://en.wikipedia.org/wiki/Integral "Integral") of non-trigonometric functions: a common technique involves first using the [substitution rule with a trigonometric function](https://en.wikipedia.org/wiki/Trigonometric_substitution "Trigonometric substitution"), and then simplifying the resulting integral with a trigonometric identity.
## Pythagorean identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=1 "Edit section: Pythagorean identities")\]
Main article: [Pythagorean trigonometric identity](https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity "Pythagorean trigonometric identity")
[](https://en.wikipedia.org/wiki/File:Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg)
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity
1
\+
cot
2
θ
\=
csc
2
θ
{\\displaystyle 1+\\cot ^{2}\\theta =\\csc ^{2}\\theta }
, and the red triangle shows that
tan
2
θ
\+
1
\=
sec
2
θ
{\\displaystyle \\tan ^{2}\\theta +1=\\sec ^{2}\\theta }
.
The basic relationship between the [sine and cosine](https://en.wikipedia.org/wiki/Sine_and_cosine "Sine and cosine") is given by the Pythagorean identity:
sin 2 θ \+ cos 2 θ \= 1 , {\\displaystyle \\sin ^{2}\\theta +\\cos ^{2}\\theta =1,} 
where sin 2 θ {\\displaystyle \\sin ^{2}\\theta }  means ( sin θ ) 2 {\\displaystyle {(\\sin \\theta )}^{2}}  and cos 2 θ {\\displaystyle \\cos ^{2}\\theta }  means ( cos θ ) 2 . {\\displaystyle {(\\cos \\theta )}^{2}.} 
This can be viewed as a version of the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"), and follows from the equation x 2 \+ y 2 \= 1 {\\displaystyle x^{2}+y^{2}=1}  for the [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle"). This equation can be solved for either the sine or the cosine:
sin θ \= ± 1 − cos 2 θ , cos θ \= ± 1 − sin 2 θ . {\\displaystyle {\\begin{aligned}\\sin \\theta &=\\pm {\\sqrt {1-\\cos ^{2}\\theta }},\\\\\\cos \\theta &=\\pm {\\sqrt {1-\\sin ^{2}\\theta }}.\\end{aligned}}} 
where the sign depends on the [quadrant](https://en.wikipedia.org/wiki/Quadrant_\(plane_geometry\) "Quadrant (plane geometry)") of θ . {\\displaystyle \\theta .} 
Dividing this identity by sin 2 θ {\\displaystyle \\sin ^{2}\\theta } , cos 2 θ {\\displaystyle \\cos ^{2}\\theta } , or both yields the following identities: 1 \+ cot 2 θ \= csc 2 θ 1 \+ tan 2 θ \= sec 2 θ sec 2 θ \+ csc 2 θ \= sec 2 θ csc 2 θ {\\displaystyle {\\begin{aligned}1+\\cot ^{2}\\theta &=\\csc ^{2}\\theta \\\\1+\\tan ^{2}\\theta &=\\sec ^{2}\\theta \\\\\\sec ^{2}\\theta +\\csc ^{2}\\theta &=\\sec ^{2}\\theta \\csc ^{2}\\theta \\end{aligned}}} 
Using these identities, it is possible to express any trigonometric function in terms of any other ([up to](https://en.wikipedia.org/wiki/Up_to "Up to") a plus or minus sign):
| in terms of | sin θ {\\displaystyle \\sin \\theta }  |
|---|---|
## Reflections, shifts, and periodicity
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=2 "Edit section: Reflections, shifts, and periodicity")\]
By examining the unit circle, one can establish the following properties of the trigonometric functions.
### Reflections
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=3 "Edit section: Reflections")\]
[](https://en.wikipedia.org/wiki/File:Unit_Circle_-_symmetry.svg)
Transformation of coordinates (*a*,*b*) when shifting the reflection angle
α
{\\displaystyle \\alpha }
in increments of
π
4
{\\displaystyle {\\frac {\\pi }{4}}}
When the direction of a [Euclidean vector](https://en.wikipedia.org/wiki/Euclidean_vector "Euclidean vector") is represented by an angle θ , {\\displaystyle \\theta ,}  this is the angle determined by the free vector (starting at the origin) and the positive x {\\displaystyle x} \-unit vector. The same concept may also be applied to lines in an [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space "Euclidean space"), where the angle is that determined by a parallel to the given line through the origin and the positive x {\\displaystyle x} \-axis. If a line (vector) with direction θ {\\displaystyle \\theta }  is reflected about a line with direction α , {\\displaystyle \\alpha ,}  then the direction angle θ ′ {\\displaystyle \\theta ^{\\prime }}  of this reflected line (vector) has the value θ ′ \= 2 α − θ . {\\displaystyle \\theta ^{\\prime }=2\\alpha -\\theta .} 
The values of the trigonometric functions of these angles θ , θ ′ {\\displaystyle \\theta ,\\;\\theta ^{\\prime }}  for specific angles α {\\displaystyle \\alpha }  satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as *reduction formulae*.[\[2\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-2)
| θ {\\displaystyle \\theta }  reflected in α \= 0 {\\displaystyle \\alpha =0} [\[3\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-3) [odd/even](https://en.wikipedia.org/wiki/Even_and_odd_functions "Even and odd functions") identities |
|---|
### Shifts and periodicity
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=4 "Edit section: Shifts and periodicity")\]
[](https://en.wikipedia.org/wiki/File:Unit_Circle_-_shifts.svg)
Transformation of coordinates (*a*,*b*) when shifting the angle
θ
{\\displaystyle \\theta }
in increments of
π
2
{\\displaystyle {\\frac {\\pi }{2}}}
| Shift by one quarter period | Shift by one half period | Shift by full periods[\[4\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-4) | Period |
|---|---|---|---|
| sin ( θ ± π 2 ) \= ± cos θ {\\displaystyle \\sin(\\theta \\pm {\\tfrac {\\pi }{2}})=\\pm \\cos \\theta }  | | | |
### Signs
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=5 "Edit section: Signs")\]
The sign of trigonometric functions depends on quadrant of the angle. If − π \< θ ≤ π {\\displaystyle {-\\pi }\<\\theta \\leq \\pi }  and sgn is the [sign function](https://en.wikipedia.org/wiki/Sign_function "Sign function"),
sgn ( sin θ ) \= sgn ( csc θ ) \= { \+ 1 if 0 \< θ \< π − 1 if − π \< θ \< 0 0 if θ ∈ { 0 , π } sgn ( cos θ ) \= sgn ( sec θ ) \= { \+ 1 if − π 2 \< θ \< π 2 − 1 if − π \< θ \< − π 2 or π 2 \< θ \< π 0 if θ ∈ { − π 2 , π 2 } sgn ( tan θ ) \= sgn ( cot θ ) \= { \+ 1 if − π \< θ \< − π 2 or 0 \< θ \< π 2 − 1 if − π 2 \< θ \< 0 or π 2 \< θ \< π 0 if θ ∈ { − π 2 , 0 , π 2 , π } {\\displaystyle {\\begin{aligned}\\operatorname {sgn}(\\sin \\theta )=\\operatorname {sgn}(\\csc \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ 0\<\\theta \<\\pi \\\\-1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<0\\\\0&{\\text{if}}\\ \\ \\theta \\in \\{0,\\pi \\}\\end{cases}}\\\\\[5mu\]\\operatorname {sgn}(\\cos \\theta )=\\operatorname {sgn}(\\sec \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ {-{\\tfrac {\\pi }{2}}}\<\\theta \<{\\tfrac {\\pi }{2}}\\\\-1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<-{\\tfrac {\\pi }{2}}\\ \\ {\\text{or}}\\ \\ {\\tfrac {\\pi }{2}}\<\\theta \<\\pi \\\\0&{\\text{if}}\\ \\ \\theta \\in {\\bigl \\{}{-{\\tfrac {\\pi }{2}}},{\\tfrac {\\pi }{2}}{\\bigr \\}}\\end{cases}}\\\\\[5mu\]\\operatorname {sgn}(\\tan \\theta )=\\operatorname {sgn}(\\cot \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<-{\\tfrac {\\pi }{2}}\\ \\ {\\text{or}}\\ \\ 0\<\\theta \<{\\tfrac {\\pi }{2}}\\\\-1&{\\text{if}}\\ \\ {-{\\tfrac {\\pi }{2}}}\<\\theta \<0\\ \\ {\\text{or}}\\ \\ {\\tfrac {\\pi }{2}}\<\\theta \<\\pi \\\\0&{\\text{if}}\\ \\ \\theta \\in {\\bigl \\{}{-{\\tfrac {\\pi }{2}}},0,{\\tfrac {\\pi }{2}},\\pi {\\bigr \\}}\\end{cases}}\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\operatorname {sgn} (\\sin \\theta )=\\operatorname {sgn} (\\csc \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ 0\<\\theta \<\\pi \\\\-1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<0\\\\0&{\\text{if}}\\ \\ \\theta \\in \\{0,\\pi \\}\\end{cases}}\\\\\[5mu\]\\operatorname {sgn} (\\cos \\theta )=\\operatorname {sgn} (\\sec \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ {-{\\tfrac {\\pi }{2}}}\<\\theta \<{\\tfrac {\\pi }{2}}\\\\-1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<-{\\tfrac {\\pi }{2}}\\ \\ {\\text{or}}\\ \\ {\\tfrac {\\pi }{2}}\<\\theta \<\\pi \\\\0&{\\text{if}}\\ \\ \\theta \\in {\\bigl \\{}{-{\\tfrac {\\pi }{2}}},{\\tfrac {\\pi }{2}}{\\bigr \\}}\\end{cases}}\\\\\[5mu\]\\operatorname {sgn} (\\tan \\theta )=\\operatorname {sgn} (\\cot \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<-{\\tfrac {\\pi }{2}}\\ \\ {\\text{or}}\\ \\ 0\<\\theta \<{\\tfrac {\\pi }{2}}\\\\-1&{\\text{if}}\\ \\ {-{\\tfrac {\\pi }{2}}}\<\\theta \<0\\ \\ {\\text{or}}\\ \\ {\\tfrac {\\pi }{2}}\<\\theta \<\\pi \\\\0&{\\text{if}}\\ \\ \\theta \\in {\\bigl \\{}{-{\\tfrac {\\pi }{2}}},0,{\\tfrac {\\pi }{2}},\\pi {\\bigr \\}}\\end{cases}}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3885aee3a70a73da0b369ad2f21a8122f4e78587)
The trigonometric functions are periodic with common period 2 π , {\\displaystyle 2\\pi ,}  so for values of θ outside the interval ( − π , π \] , {\\displaystyle ({-\\pi },\\pi \],} ![{\\displaystyle ({-\\pi },\\pi \],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/122328145a5f3a4162d21b4fbb5f4d1149932d2b) they take repeating values (see [§ Shifts and periodicity](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Shifts_and_periodicity) above). The sign of a sinusoid or cosinusoid can be used to define a normalized [square wave](https://en.wikipedia.org/wiki/Square_wave_\(waveform\) "Square wave (waveform)"). For example, the functions sgn ( sin x ) {\\displaystyle \\operatorname {sgn}(\\sin x)}  and sgn ( cos x ) {\\displaystyle \\operatorname {sgn}(\\cos x)}  take values ±1 and correspond to square waves with a phase shift of *π*/2.
## Angle sum and difference identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=6 "Edit section: Angle sum and difference identities")\]
See also: [Proofs of trigonometric identities § Angle sum identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Angle_sum_identities "Proofs of trigonometric identities"), and [Small-angle approximation § Angle sum and difference](https://en.wikipedia.org/wiki/Small-angle_approximation#Angle_sum_and_difference "Small-angle approximation")
[](https://en.wikipedia.org/wiki/File:Angle_sum.svg)
Geometric construction to derive angle sum trigonometric identities
[](https://en.wikipedia.org/wiki/File:Diagram_showing_the_angle_difference_trigonometry_identities_for_sin\(a-b\)_and_cos\(a-b\).svg)
Diagram showing the angle difference identities for
sin
(
α
−
β
)
{\\displaystyle \\sin(\\alpha -\\beta )}
and
cos
(
α
−
β
)
{\\displaystyle \\cos(\\alpha -\\beta )}
These are also known as the *angle addition and subtraction theorems* (or *formulae*). sin ( α \+ β ) \= sin α cos β \+ cos α sin β sin ( α − β ) \= sin α cos β − cos α sin β cos ( α \+ β ) \= cos α cos β − sin α sin β cos ( α − β ) \= cos α cos β \+ sin α sin β {\\displaystyle {\\begin{aligned}\\sin(\\alpha +\\beta )&=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta \\\\\\sin(\\alpha -\\beta )&=\\sin \\alpha \\cos \\beta -\\cos \\alpha \\sin \\beta \\\\\\cos(\\alpha +\\beta )&=\\cos \\alpha \\cos \\beta -\\sin \\alpha \\sin \\beta \\\\\\cos(\\alpha -\\beta )&=\\cos \\alpha \\cos \\beta +\\sin \\alpha \\sin \\beta \\end{aligned}}} 
The angle difference identities for sin ( α − β ) {\\displaystyle \\sin(\\alpha -\\beta )}  and cos ( α − β ) {\\displaystyle \\cos(\\alpha -\\beta )}  can be derived from the angle sum versions (and vice versa) by substituting − β {\\displaystyle -\\beta }  for β {\\displaystyle \\beta }  and using the facts that sin ( − β ) \= − sin ( β ) {\\displaystyle \\sin(-\\beta )=-\\sin(\\beta )}  and cos ( − β ) \= cos ( β ) {\\displaystyle \\cos(-\\beta )=\\cos(\\beta )}  They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here. They can also be seen as expressing the [dot product](https://en.wikipedia.org/wiki/Dot_product "Dot product") and [cross product](https://en.wikipedia.org/wiki/Cross_product "Cross product") of two vectors in terms of the cosine and the sine of the angle between them.
These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.
| | |
|---|---|
| Sine | sin ( α ± β ) {\\displaystyle \\sin(\\alpha \\pm \\beta )}  |
### Sines and cosines of sums of infinitely many angles
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=7 "Edit section: Sines and cosines of sums of infinitely many angles")\]
When the series ∑ i \= 1 ∞ θ i {\\textstyle \\sum \_{i=1}^{\\infty }\\theta \_{i}}  [converges absolutely](https://en.wikipedia.org/wiki/Absolute_convergence "Absolute convergence") then
sin ( ∑ i \= 1 ∞ θ i ) \= ∑ odd k ≥ 1 ( − 1 ) k − 1 2 ∑ A ⊆ { 1 , 2 , 3 , … } \| A \| \= k ( ∏ i ∈ A sin θ i ∏ i ∉ A cos θ i ) cos ( ∑ i \= 1 ∞ θ i ) \= ∑ even k ≥ 0 ( − 1 ) k 2 ∑ A ⊆ { 1 , 2 , 3 , … } \| A \| \= k ( ∏ i ∈ A sin θ i ∏ i ∉ A cos θ i ) . {\\displaystyle {\\begin{aligned}{\\sin }{\\biggl (}\\sum \_{i=1}^{\\infty }\\theta \_{i}{\\biggl )}&=\\sum \_{{\\text{odd}}\\ k\\geq 1}(-1)^{\\frac {k-1}{2}}\\!\\!\\sum \_{\\begin{smallmatrix}A\\subseteq \\{\\,1,2,3,\\dots \\,\\}\\\\\\left\|A\\right\|=k\\end{smallmatrix}}{\\biggl (}\\prod \_{i\\in A}\\sin \\theta \_{i}\\prod \_{i\\not \\in A}\\cos \\theta \_{i}{\\biggr )}\\\\{\\cos }{\\biggl (}\\sum \_{i=1}^{\\infty }\\theta \_{i}{\\biggr )}&=\\sum \_{{\\text{even}}\\ k\\geq 0}(-1)^{\\frac {k}{2}}\\,\\sum \_{\\begin{smallmatrix}A\\subseteq \\{\\,1,2,3,\\dots \\,\\}\\\\\\left\|A\\right\|=k\\end{smallmatrix}}{\\biggl (}\\prod \_{i\\in A}\\sin \\theta \_{i}\\prod \_{i\\not \\in A}\\cos \\theta \_{i}{\\biggr )}.\\end{aligned}}} 
Because the series ∑ i \= 1 ∞ θ i {\\textstyle \\sum \_{i=1}^{\\infty }\\theta \_{i}}  converges absolutely, it is necessarily the case that lim i → ∞ θ i \= 0 , {\\textstyle \\lim \_{i\\to \\infty }\\theta \_{i}=0,}  lim i → ∞ sin θ i \= 0 , {\\textstyle \\lim \_{i\\to \\infty }\\sin \\theta \_{i}=0,}  and lim i → ∞ cos θ i \= 1\. {\\textstyle \\lim \_{i\\to \\infty }\\cos \\theta \_{i}=1.}  Particularly, in these two identities, an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are [cofinitely](https://en.wikipedia.org/wiki/Cofiniteness "Cofiniteness") many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.
When only finitely many of the angles θ i {\\displaystyle \\theta \_{i}}  are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
### Tangents and cotangents of sums
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=8 "Edit section: Tangents and cotangents of sums")\]
Let e k {\\displaystyle e\_{k}}  (for k \= 0 , 1 , 2 , 3 , … {\\displaystyle k=0,1,2,3,\\ldots } ) be the kth-degree [elementary symmetric polynomial](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial "Elementary symmetric polynomial") in the variables x i \= tan θ i {\\displaystyle x\_{i}=\\tan \\theta \_{i}}  for i \= 0 , 1 , 2 , 3 , … , {\\displaystyle i=0,1,2,3,\\ldots ,}  that is,
e 0 \= 1 e 1 \= ∑ i x i \= ∑ i tan θ i e 2 \= ∑ i \< j x i x j \= ∑ i \< j tan θ i tan θ j e 3 \= ∑ i \< j \< k x i x j x k \= ∑ i \< j \< k tan θ i tan θ j tan θ k ⋮ ⋮ {\\displaystyle {\\begin{aligned}e\_{0}&=1\\\\\[6pt\]e\_{1}&=\\sum \_{i}x\_{i}&&=\\sum \_{i}\\tan \\theta \_{i}\\\\\[6pt\]e\_{2}&=\\sum \_{i\<j}x\_{i}x\_{j}&&=\\sum \_{i\<j}\\tan \\theta \_{i}\\tan \\theta \_{j}\\\\\[6pt\]e\_{3}&=\\sum \_{i\<j\<k}x\_{i}x\_{j}x\_{k}&&=\\sum \_{i\<j\<k}\\tan \\theta \_{i}\\tan \\theta \_{j}\\tan \\theta \_{k}\\\\&\\ \\ \\vdots &&\\ \\ \\vdots \\end{aligned}}} ![{\\displaystyle {\\begin{aligned}e\_{0}&=1\\\\\[6pt\]e\_{1}&=\\sum \_{i}x\_{i}&&=\\sum \_{i}\\tan \\theta \_{i}\\\\\[6pt\]e\_{2}&=\\sum \_{i\<j}x\_{i}x\_{j}&&=\\sum \_{i\<j}\\tan \\theta \_{i}\\tan \\theta \_{j}\\\\\[6pt\]e\_{3}&=\\sum \_{i\<j\<k}x\_{i}x\_{j}x\_{k}&&=\\sum \_{i\<j\<k}\\tan \\theta \_{i}\\tan \\theta \_{j}\\tan \\theta \_{k}\\\\&\\ \\ \\vdots &&\\ \\ \\vdots \\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1927ae2be96f5156e18813a20486fdb0ff72f1)
Then
tan ( ∑ i θ i ) \= e 1 − e 3 \+ e 5 − ⋯ e 0 − e 2 \+ e 4 − ⋯ . {\\displaystyle \\tan {\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}={\\frac {e\_{1}-e\_{3}+e\_{5}-\\cdots }{e\_{0}-e\_{2}+e\_{4}-\\cdots }}.}  This can be shown by using the sine and cosine sum formulae above: tan ( ∑ i θ i ) \= sin ( ∑ i θ i ) / ∏ i cos θ i cos ( ∑ i θ i ) / ∏ i cos θ i \= ∑ odd k ≥ 1 ( − 1 ) k − 1 2 ∑ A ⊆ { 1 , 2 , 3 , … } \| A \| \= k ∏ i ∈ A tan θ i ∑ even k ≥ 0 ( − 1 ) k 2 ∑ A ⊆ { 1 , 2 , 3 , … } \| A \| \= k ∏ i ∈ A tan θ i \= e 1 − e 3 \+ e 5 − ⋯ e 0 − e 2 \+ e 4 − ⋯ cot ( ∑ i θ i ) \= e 0 − e 2 \+ e 4 − ⋯ e 1 − e 3 \+ e 5 − ⋯ {\\displaystyle {\\begin{aligned}\\tan {\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {{\\sin }{\\bigl (}\\sum \_{i}\\theta \_{i}{\\bigr )}/\\prod \_{i}\\cos \\theta \_{i}}{{\\cos }{\\bigl (}\\sum \_{i}\\theta \_{i}{\\bigr )}/\\prod \_{i}\\cos \\theta \_{i}}}\\\\\[10pt\]&={\\frac {\\displaystyle \\sum \_{{\\text{odd}}\\ k\\geq 1}(-1)^{\\frac {k-1}{2}}\\sum \_{\\begin{smallmatrix}A\\subseteq \\{1,2,3,\\dots \\}\\\\\\left\|A\\right\|=k\\end{smallmatrix}}\\prod \_{i\\in A}\\tan \\theta \_{i}}{\\displaystyle \\sum \_{{\\text{even}}\\ k\\geq 0}~(-1)^{\\frac {k}{2}}\~~\\sum \_{\\begin{smallmatrix}A\\subseteq \\{1,2,3,\\dots \\}\\\\\\left\|A\\right\|=k\\end{smallmatrix}}\\prod \_{i\\in A}\\tan \\theta \_{i}}}={\\frac {e\_{1}-e\_{3}+e\_{5}-\\cdots }{e\_{0}-e\_{2}+e\_{4}-\\cdots }}\\\\\[10pt\]\\cot {\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {e\_{0}-e\_{2}+e\_{4}-\\cdots }{e\_{1}-e\_{3}+e\_{5}-\\cdots }}\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\tan {\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {{\\sin }{\\bigl (}\\sum \_{i}\\theta \_{i}{\\bigr )}/\\prod \_{i}\\cos \\theta \_{i}}{{\\cos }{\\bigl (}\\sum \_{i}\\theta \_{i}{\\bigr )}/\\prod \_{i}\\cos \\theta \_{i}}}\\\\\[10pt\]&={\\frac {\\displaystyle \\sum \_{{\\text{odd}}\\ k\\geq 1}(-1)^{\\frac {k-1}{2}}\\sum \_{\\begin{smallmatrix}A\\subseteq \\{1,2,3,\\dots \\}\\\\\\left\|A\\right\|=k\\end{smallmatrix}}\\prod \_{i\\in A}\\tan \\theta \_{i}}{\\displaystyle \\sum \_{{\\text{even}}\\ k\\geq 0}~(-1)^{\\frac {k}{2}}\~~\\sum \_{\\begin{smallmatrix}A\\subseteq \\{1,2,3,\\dots \\}\\\\\\left\|A\\right\|=k\\end{smallmatrix}}\\prod \_{i\\in A}\\tan \\theta \_{i}}}={\\frac {e\_{1}-e\_{3}+e\_{5}-\\cdots }{e\_{0}-e\_{2}+e\_{4}-\\cdots }}\\\\\[10pt\]\\cot {\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {e\_{0}-e\_{2}+e\_{4}-\\cdots }{e\_{1}-e\_{3}+e\_{5}-\\cdots }}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ff3a24e1e073c731c170651eca41efdad72212)
The number of terms on the right side depends on the number of terms on the left side.
For example: tan ( θ 1 \+ θ 2 ) \= e 1 e 0 − e 2 \= x 1 \+ x 2 1 − x 1 x 2 \= tan θ 1 \+ tan θ 2 1 − tan θ 1 tan θ 2 , tan ( θ 1 \+ θ 2 \+ θ 3 ) \= e 1 − e 3 e 0 − e 2 \= ( x 1 \+ x 2 \+ x 3 ) − ( x 1 x 2 x 3 ) 1 − ( x 1 x 2 \+ x 1 x 3 \+ x 2 x 3 ) , tan ( θ 1 \+ θ 2 \+ θ 3 \+ θ 4 ) \= e 1 − e 3 e 0 − e 2 \+ e 4 \= ( x 1 \+ x 2 \+ x 3 \+ x 4 ) − ( x 1 x 2 x 3 \+ x 1 x 2 x 4 \+ x 1 x 3 x 4 \+ x 2 x 3 x 4 ) 1 − ( x 1 x 2 \+ x 1 x 3 \+ x 1 x 4 \+ x 2 x 3 \+ x 2 x 4 \+ x 3 x 4 ) \+ ( x 1 x 2 x 3 x 4 ) , {\\displaystyle {\\begin{aligned}\\tan(\\theta \_{1}+\\theta \_{2})&={\\frac {e\_{1}}{e\_{0}-e\_{2}}}={\\frac {x\_{1}+x\_{2}}{1\\ -\\ x\_{1}x\_{2}}}={\\frac {\\tan \\theta \_{1}+\\tan \\theta \_{2}}{1\\ -\\ \\tan \\theta \_{1}\\tan \\theta \_{2}}},\\\\\[8pt\]\\tan(\\theta \_{1}+\\theta \_{2}+\\theta \_{3})&={\\frac {e\_{1}-e\_{3}}{e\_{0}-e\_{2}}}={\\frac {(x\_{1}+x\_{2}+x\_{3})\\ -\\ (x\_{1}x\_{2}x\_{3})}{1\\ -\\ (x\_{1}x\_{2}+x\_{1}x\_{3}+x\_{2}x\_{3})}},\\\\\[8pt\]\\tan(\\theta \_{1}+\\theta \_{2}+\\theta \_{3}+\\theta \_{4})&={\\frac {e\_{1}-e\_{3}}{e\_{0}-e\_{2}+e\_{4}}}\\\\\[8pt\]&={\\frac {(x\_{1}+x\_{2}+x\_{3}+x\_{4})\\ -\\ (x\_{1}x\_{2}x\_{3}+x\_{1}x\_{2}x\_{4}+x\_{1}x\_{3}x\_{4}+x\_{2}x\_{3}x\_{4})}{1\\ -\\ (x\_{1}x\_{2}+x\_{1}x\_{3}+x\_{1}x\_{4}+x\_{2}x\_{3}+x\_{2}x\_{4}+x\_{3}x\_{4})\\ +\\ (x\_{1}x\_{2}x\_{3}x\_{4})}},\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\tan(\\theta \_{1}+\\theta \_{2})&={\\frac {e\_{1}}{e\_{0}-e\_{2}}}={\\frac {x\_{1}+x\_{2}}{1\\ -\\ x\_{1}x\_{2}}}={\\frac {\\tan \\theta \_{1}+\\tan \\theta \_{2}}{1\\ -\\ \\tan \\theta \_{1}\\tan \\theta \_{2}}},\\\\\[8pt\]\\tan(\\theta \_{1}+\\theta \_{2}+\\theta \_{3})&={\\frac {e\_{1}-e\_{3}}{e\_{0}-e\_{2}}}={\\frac {(x\_{1}+x\_{2}+x\_{3})\\ -\\ (x\_{1}x\_{2}x\_{3})}{1\\ -\\ (x\_{1}x\_{2}+x\_{1}x\_{3}+x\_{2}x\_{3})}},\\\\\[8pt\]\\tan(\\theta \_{1}+\\theta \_{2}+\\theta \_{3}+\\theta \_{4})&={\\frac {e\_{1}-e\_{3}}{e\_{0}-e\_{2}+e\_{4}}}\\\\\[8pt\]&={\\frac {(x\_{1}+x\_{2}+x\_{3}+x\_{4})\\ -\\ (x\_{1}x\_{2}x\_{3}+x\_{1}x\_{2}x\_{4}+x\_{1}x\_{3}x\_{4}+x\_{2}x\_{3}x\_{4})}{1\\ -\\ (x\_{1}x\_{2}+x\_{1}x\_{3}+x\_{1}x\_{4}+x\_{2}x\_{3}+x\_{2}x\_{4}+x\_{3}x\_{4})\\ +\\ (x\_{1}x\_{2}x\_{3}x\_{4})}},\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9196b8ab5e0aef6cf7d7fec7d7546f2a6e7d2f)
and so on. The case of only finitely many terms can be proved by [mathematical induction](https://en.wikipedia.org/wiki/Mathematical_induction "Mathematical induction").[\[14\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-14) The case of infinitely many terms can be proved by using some elementary inequalities.[\[15\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-15)
### Linear fractional transformations of tangents, related to tangents of sums
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=9 "Edit section: Linear fractional transformations of tangents, related to tangents of sums")\]
Suppose a , b , c , d , p , q ∈ R {\\textstyle a,b,c,d,p,q\\in \\mathbb {R} }  and i \= − 1 {\\textstyle i={\\sqrt {-1}}}  and
a
i
\+
b
c
i
\+
d
\=
p
i
\+
q
{\\displaystyle {\\frac {ai+b}{ci+d}}=pi+q}
and let φ {\\textstyle \\varphi }  be any number for which tan φ \= c d . {\\textstyle \\tan \\varphi ={\\tfrac {c}{d}}.}  Suppose that a c ≠ b d {\\textstyle {\\tfrac {a}{c}}\\neq {\\tfrac {b}{d}}}  so that the forgoing fraction cannot be 0/0. Then for all θ ∈ R {\\textstyle \\theta \\in \\mathbb {R} } [\[16\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-16)
a
tan
θ
\+
b
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tan
θ
\+
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\=
p
tan
(
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−
φ
)
\+
q
.
{\\displaystyle {\\frac {a\\tan \\theta +b}{c\\tan \\theta +d}}=p\\tan(\\theta -\\varphi )+q.}
(In case the denominator of this fraction is 0, we take the value of the fraction to be ∞ {\\textstyle \\infty } , where the symbol ∞ {\\textstyle \\infty }  does not mean either \+ ∞ {\\textstyle +\\infty }  or − ∞ {\\textstyle -\\infty } , but is the ∞ {\\textstyle \\infty }  that is approached by going in either the positive or the negative direction, making the completion of the line R ∪ { ∞ } {\\textstyle \\mathbb {R} \\cup \\{\\,\\infty \\,\\}}  topologically a circle.)
From this identity it can be shown to follow quickly that the family of all [Cauchy-distributed](https://en.wikipedia.org/wiki/Cauchy_distribution "Cauchy distribution") random variables is closed under linear fractional transformations, a result known since 1976.[\[17\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-17)
### Secants and cosecants of sums
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=10 "Edit section: Secants and cosecants of sums")\]
sec ( ∑ i θ i ) \= ∏ i sec θ i e 0 − e 2 \+ e 4 − ⋯ csc ( ∑ i θ i ) \= ∏ i sec θ i e 1 − e 3 \+ e 5 − ⋯ {\\displaystyle {\\begin{aligned}{\\sec }{\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {\\prod \_{i}\\sec \\theta \_{i}}{e\_{0}-e\_{2}+e\_{4}-\\cdots }}\\\\\[8pt\]{\\csc }{\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {\\prod \_{i}\\sec \\theta \_{i}}{e\_{1}-e\_{3}+e\_{5}-\\cdots }}\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}{\\sec }{\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {\\prod \_{i}\\sec \\theta \_{i}}{e\_{0}-e\_{2}+e\_{4}-\\cdots }}\\\\\[8pt\]{\\csc }{\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {\\prod \_{i}\\sec \\theta \_{i}}{e\_{1}-e\_{3}+e\_{5}-\\cdots }}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b24f2b169fbe11608507c63f5bb728e03b2b4957)
where e k {\\displaystyle e\_{k}}  is the kth-degree [elementary symmetric polynomial](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial "Elementary symmetric polynomial") in the n variables x i \= tan θ i , {\\displaystyle x\_{i}=\\tan \\theta \_{i},}  i \= 1 , … , n , {\\displaystyle i=1,\\ldots ,n,}  and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.[\[18\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-18) The case of only finitely many terms can be proved by mathematical induction on the number of such terms.
For example,
sec ( α \+ β \+ γ ) \= sec α sec β sec γ 1 − tan α tan β − tan α tan γ − tan β tan γ csc ( α \+ β \+ γ ) \= sec α sec β sec γ tan α \+ tan β \+ tan γ − tan α tan β tan γ . {\\displaystyle {\\begin{aligned}\\sec(\\alpha +\\beta +\\gamma )&={\\frac {\\sec \\alpha \\sec \\beta \\sec \\gamma }{1-\\tan \\alpha \\tan \\beta -\\tan \\alpha \\tan \\gamma -\\tan \\beta \\tan \\gamma }}\\\\\[8pt\]\\csc(\\alpha +\\beta +\\gamma )&={\\frac {\\sec \\alpha \\sec \\beta \\sec \\gamma }{\\tan \\alpha +\\tan \\beta +\\tan \\gamma -\\tan \\alpha \\tan \\beta \\tan \\gamma }}.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\sec(\\alpha +\\beta +\\gamma )&={\\frac {\\sec \\alpha \\sec \\beta \\sec \\gamma }{1-\\tan \\alpha \\tan \\beta -\\tan \\alpha \\tan \\gamma -\\tan \\beta \\tan \\gamma }}\\\\\[8pt\]\\csc(\\alpha +\\beta +\\gamma )&={\\frac {\\sec \\alpha \\sec \\beta \\sec \\gamma }{\\tan \\alpha +\\tan \\beta +\\tan \\gamma -\\tan \\alpha \\tan \\beta \\tan \\gamma }}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e082e69aab7a9c782f590f3a93262ba3101899b7)
### Ptolemy's theorem
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=11 "Edit section: Ptolemy's theorem")\]
Main article: [Ptolemy's theorem](https://en.wikipedia.org/wiki/Ptolemy%27s_theorem "Ptolemy's theorem")
See also: [History of trigonometry § Classical antiquity](https://en.wikipedia.org/wiki/History_of_trigonometry#Classical_antiquity "History of trigonometry")
[](https://en.wikipedia.org/wiki/File:Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg)
Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine. Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(*α* + *β*) = sin *α* cos *β* + cos *α* sin *β*.
Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a [cyclic quadrilateral](https://en.wikipedia.org/wiki/Cyclic_quadrilateral "Cyclic quadrilateral") A B C D {\\displaystyle ABCD} , as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.[\[19\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-cut-the-knot.org-19) The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.
By [Thales's theorem](https://en.wikipedia.org/wiki/Thales%27s_theorem "Thales's theorem"), ∠ D A B {\\displaystyle \\angle DAB}  and ∠ D C B {\\displaystyle \\angle DCB}  are both right angles. The right-angled triangles D A B {\\displaystyle DAB}  and D C B {\\displaystyle DCB}  both share the hypotenuse B D ¯ {\\displaystyle {\\overline {BD}}}  of length 1. Thus, the side A B ¯ \= sin α {\\displaystyle {\\overline {AB}}=\\sin \\alpha } , A D ¯ \= cos α {\\displaystyle {\\overline {AD}}=\\cos \\alpha } , B C ¯ \= sin β {\\displaystyle {\\overline {BC}}=\\sin \\beta }  and C D ¯ \= cos β {\\displaystyle {\\overline {CD}}=\\cos \\beta } .
By the [inscribed angle](https://en.wikipedia.org/wiki/Inscribed_angle "Inscribed angle") theorem, the [central angle](https://en.wikipedia.org/wiki/Central_angle "Central angle") subtended by the chord A C ¯ {\\displaystyle {\\overline {AC}}}  at the circle's center is twice the angle ∠ A D C {\\displaystyle \\angle ADC} , i.e. 2 ( α \+ β ) {\\displaystyle 2(\\alpha +\\beta )} . Therefore, the symmetrical pair of red triangles each has the angle α \+ β {\\displaystyle \\alpha +\\beta }  at the center. Each of these triangles has a [hypotenuse](https://en.wikipedia.org/wiki/Hypotenuse "Hypotenuse") of length 1 2 {\\textstyle {\\frac {1}{2}}} , so the length of A C ¯ {\\displaystyle {\\overline {AC}}}  is 2 × 1 2 sin ( α \+ β ) {\\textstyle 2\\times {\\frac {1}{2}}\\sin(\\alpha +\\beta )} , i.e. simply sin ( α \+ β ) {\\displaystyle \\sin(\\alpha +\\beta )} . The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also sin ( α \+ β ) {\\displaystyle \\sin(\\alpha +\\beta )} .
When these values are substituted into the statement of Ptolemy's theorem that \| A C ¯ \| ⋅ \| B D ¯ \| \= \| A B ¯ \| ⋅ \| C D ¯ \| \+ \| A D ¯ \| ⋅ \| B C ¯ \| {\\displaystyle \|{\\overline {AC}}\|\\cdot \|{\\overline {BD}}\|=\|{\\overline {AB}}\|\\cdot \|{\\overline {CD}}\|+\|{\\overline {AD}}\|\\cdot \|{\\overline {BC}}\|} , this yields the angle sum trigonometric identity for sine: sin ( α \+ β ) \= sin α cos β \+ cos α sin β {\\displaystyle \\sin(\\alpha +\\beta )=\\sin \\alpha \\cos \\beta +\\cos \\alpha \\sin \\beta } . The angle difference formula for sin ( α − β ) {\\displaystyle \\sin(\\alpha -\\beta )}  can be similarly derived by letting the side C D ¯ {\\displaystyle {\\overline {CD}}}  serve as a diameter instead of B D ¯ {\\displaystyle {\\overline {BD}}} .[\[19\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-cut-the-knot.org-19)
## Multiple-angle and half-angle formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=12 "Edit section: Multiple-angle and half-angle formulas")\]
| | |
|---|---|
| Tn is the nth [Chebyshev polynomial](https://en.wikipedia.org/wiki/Chebyshev_polynomials "Chebyshev polynomials") | cos ( n θ ) \= T n ( cos θ ) {\\displaystyle \\cos(n\\theta )=T\_{n}(\\cos \\theta )} [\[20\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-20) |
### Multiple-angle formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=13 "Edit section: Multiple-angle formulae")\]
#### Double-angle formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=14 "Edit section: Double-angle formulae")\]
[](https://en.wikipedia.org/wiki/File:Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg)
Visual demonstration of the double-angle formula for sine. For the above isosceles triangle with unit sides and angle
2
θ
{\\displaystyle 2\\theta }
, the area
1/2 × base × height is calculated in two orientations. When upright, the area is
sin
θ
cos
θ
{\\displaystyle \\sin \\theta \\cos \\theta }
. When on its side, the same area is
1
2
sin
2
θ
{\\textstyle {\\tfrac {1}{2}}\\sin 2\\theta }
. Therefore,
sin
2
θ
\=
2
sin
θ
cos
θ
.
{\\displaystyle \\sin 2\\theta =2\\sin \\theta \\cos \\theta .}
Formulae for twice an angle.[\[22\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-STM1-22)[\[23\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-A&S-23)
sin ( 2 θ ) \= 2 sin θ cos θ \= ( sin θ \+ cos θ ) 2 − 1 \= 2 tan θ 1 \+ tan 2 θ cos ( 2 θ ) \= cos 2 θ − sin 2 θ \= 2 cos 2 θ − 1 \= 1 − 2 sin 2 θ \= 1 − tan 2 θ 1 \+ tan 2 θ tan ( 2 θ ) \= 2 tan θ 1 − tan 2 θ cot ( 2 θ ) \= cot 2 θ − 1 2 cot θ \= 1 − tan 2 θ 2 tan θ sec ( 2 θ ) \= sec 2 θ 2 − sec 2 θ \= 1 \+ tan 2 θ 1 − tan 2 θ csc ( 2 θ ) \= sec θ csc θ 2 \= 1 \+ tan 2 θ 2 tan θ {\\displaystyle {\\begin{aligned}\\sin(2\\theta )&=2\\sin \\theta \\cos \\theta &&=(\\sin \\theta +\\cos \\theta )^{2}-1&&&={\\frac {2\\tan \\theta }{1+\\tan ^{2}\\theta }}\\\\\\cos(2\\theta )&=\\cos ^{2}\\theta -\\sin ^{2}\\theta =2\\cos ^{2}\\theta -1&&=1-2\\sin ^{2}\\theta &&&={\\frac {1-\\tan ^{2}\\theta }{1+\\tan ^{2}\\theta }}\\\\\\tan(2\\theta )&={\\frac {2\\tan \\theta }{1-\\tan ^{2}\\theta }}\\\\\\cot(2\\theta )&={\\frac {\\cot ^{2}\\theta -1}{2\\cot \\theta }}&&={\\frac {1-\\tan ^{2}\\theta }{2\\tan \\theta }}\\\\\\sec(2\\theta )&={\\frac {\\sec ^{2}\\theta }{2-\\sec ^{2}\\theta }}&&={\\frac {1+\\tan ^{2}\\theta }{1-\\tan ^{2}\\theta }}\\\\\\csc(2\\theta )&={\\frac {\\sec \\theta \\csc \\theta }{2}}&&={\\frac {1+\\tan ^{2}\\theta }{2\\tan \\theta }}\\\\\\end{aligned}}} 
#### Triple-angle formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=15 "Edit section: Triple-angle formulae")\]
Formulae for triple angles.[\[22\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-STM1-22)
sin ( 3 θ ) \= 3 sin θ − 4 sin 3 θ \= 4 sin θ sin ( π 3 − θ ) sin ( π 3 \+ θ ) cos ( 3 θ ) \= 4 cos 3 θ − 3 cos θ \= 4 cos θ cos ( π 3 − θ ) cos ( π 3 \+ θ ) tan ( 3 θ ) \= 3 tan θ − tan 3 θ 1 − 3 tan 2 θ \= tan θ tan ( π 3 − θ ) tan ( π 3 \+ θ ) cot ( 3 θ ) \= 3 cot θ − cot 3 θ 1 − 3 cot 2 θ sec ( 3 θ ) \= sec 3 θ 4 − 3 sec 2 θ csc ( 3 θ ) \= csc 3 θ 3 csc 2 θ − 4 {\\displaystyle {\\begin{aligned}\\sin(3\\theta )&=3\\sin \\theta -4\\sin ^{3}\\theta &&=4\\sin \\theta \\sin \\left({\\tfrac {\\pi }{3}}-\\theta \\right)\\sin \\left({\\tfrac {\\pi }{3}}+\\theta \\right)\\\\\\cos(3\\theta )&=4\\cos ^{3}\\theta -3\\cos \\theta &&=4\\cos \\theta \\cos \\left({\\tfrac {\\pi }{3}}-\\theta \\right)\\cos \\left({\\tfrac {\\pi }{3}}+\\theta \\right)\\\\\\tan(3\\theta )&={\\frac {3\\tan \\theta -\\tan ^{3}\\theta }{1-3\\tan ^{2}\\theta }}&&=\\tan \\theta \\tan \\left({\\tfrac {\\pi }{3}}-\\theta \\right)\\tan \\left({\\tfrac {\\pi }{3}}+\\theta \\right)\\\\\\cot(3\\theta )&={\\frac {3\\cot \\theta -\\cot ^{3}\\theta }{1-3\\cot ^{2}\\theta }}\\\\\\sec(3\\theta )&={\\frac {\\sec ^{3}\\theta }{4-3\\sec ^{2}\\theta }}\\\\\\csc(3\\theta )&={\\frac {\\csc ^{3}\\theta }{3\\csc ^{2}\\theta -4}}\\\\\\end{aligned}}} 
#### Multiple-angle formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=16 "Edit section: Multiple-angle formulae")\]
Formulae for multiple angles.[\[24\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-24)
sin ( n θ ) \= ∑ k ∈ O \+ n ( − 1 ) k − 1 2 ( n k ) cos n − k θ sin k θ \= sin θ ∑ i \= 0 n \+ 1 2 ∑ j \= 0 i ( − 1 ) i − j ( n 2 i \+ 1 ) ( i j ) cos n − 2 ( i − j ) − 1 θ \= sin ( θ ) ∑ k \= 0 ⌊ n − 1 2 ⌋ ( − 1 ) k ( 2 cos ( θ ) ) n − 2 k − 1 ( n − k − 1 k ) \= 2 ( n − 1 ) ∏ k \= 0 n − 1 sin ( k π n \+ θ ) cos ( n θ ) \= ∑ k ∈ E 0 \+ n ( − 1 ) k 2 ( n k ) cos n − k θ sin k θ \= ∑ i \= 0 n 2 ∑ j \= 0 i ( − 1 ) i − j ( n 2 i ) ( i j ) cos n − 2 ( i − j ) θ \= ∑ k \= 0 ⌊ n 2 ⌋ ( − 1 ) k ( 2 cos ( θ ) ) n − 2 k ( n − k k ) n 2 n − 2 k cos ( ( 2 n \+ 1 ) θ ) \= ( − 1 ) n 2 2 n ∏ k \= 0 2 n cos ( k π 2 n \+ 1 − θ ) cos ( 2 n θ ) \= ( − 1 ) n 2 2 n − 1 ∏ k \= 0 2 n − 1 cos ( ( 1 \+ 2 k ) π 4 n − θ ) tan ( n θ ) \= ∑ k ∈ O \+ n ( − 1 ) k − 1 2 ( n k ) tan k θ ∑ k ∈ E 0 \+ n ( − 1 ) k 2 ( n k ) tan k θ O \+ \= Positive odd integers E 0 \+ \= Non-negative even integers {\\displaystyle {\\begin{aligned}\\sin(n\\theta )&=\\sum \_{k\\in \\mathbb {O} ^{+}}^{n}(-1)^{\\frac {k-1}{2}}{n \\choose k}\\cos ^{n-k}\\theta \\sin ^{k}\\theta =\\sin \\theta \\sum \_{i=0}^{\\frac {n+1}{2}}\\sum \_{j=0}^{i}(-1)^{i-j}{n \\choose 2i+1}{i \\choose j}\\cos ^{n-2(i-j)-1}\\theta \\\\&=\\sin(\\theta )\\sum \_{k=0}^{\\left\\lfloor {\\frac {n-1}{2}}\\right\\rfloor }(-1)^{k}{\\bigl (}2\\cos(\\theta ){\\bigr )}^{n-2k-1}{n-k-1 \\choose k}\\\\&=2^{(n-1)}\\prod \_{k=0}^{n-1}\\sin \\left({\\frac {k\\pi }{n}}+\\theta \\right)\\\\\\cos(n\\theta )&=\\sum \_{k\\in \\mathbb {E} \_{0}^{+}}^{n}(-1)^{\\frac {k}{2}}{n \\choose k}\\cos ^{n-k}\\theta \\sin ^{k}\\theta =\\sum \_{i=0}^{\\frac {n}{2}}\\sum \_{j=0}^{i}(-1)^{i-j}{n \\choose 2i}{i \\choose j}\\cos ^{n-2(i-j)}\\theta \\\\&=\\sum \_{k=0}^{\\left\\lfloor {\\frac {n}{2}}\\right\\rfloor }(-1)^{k}{(2\\cos(\\theta ))}^{n-2k}{n-k \\choose k}{\\frac {n}{2n-2k}}\\\\\\cos {\\bigl (}(2n+1)\\theta {\\bigr )}&=(-1)^{n}2^{2n}\\prod \_{k=0}^{2n}\\cos \\left({\\frac {k\\pi }{2n+1}}-\\theta \\right)\\\\\\cos(2n\\theta )&=(-1)^{n}2^{2n-1}\\prod \_{k=0}^{2n-1}\\cos \\left({\\frac {(1+2k)\\pi }{4n}}-\\theta \\right)\\\\\\tan(n\\theta )&={\\frac {\\displaystyle \\sum \_{k\\in \\mathbb {O} ^{+}}^{n}(-1)^{\\frac {k-1}{2}}{n \\choose k}\\tan ^{k}\\theta }{\\displaystyle \\sum \_{k\\in \\mathbb {E} \_{0}^{+}}^{n}(-1)^{\\frac {k}{2}}{n \\choose k}\\tan ^{k}\\theta }}\\\\\\mathbb {O} ^{+}&={\\text{Positive odd integers}}\\\\\\mathbb {E} \_{0}^{+}&={\\text{Non-negative even integers}}\\\\\\end{aligned}}} 
#### Chebyshev method
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=17 "Edit section: Chebyshev method")\]
The [Chebyshev](https://en.wikipedia.org/wiki/Pafnuty_Chebyshev "Pafnuty Chebyshev") method is a [recursive](https://en.wikipedia.org/wiki/Recursion "Recursion") [algorithm](https://en.wikipedia.org/wiki/Algorithm "Algorithm") for finding the nth multiple angle formula knowing the ( n − 1 ) {\\displaystyle (n-1)} th and ( n − 2 ) {\\displaystyle (n-2)} th values.[\[25\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-25)
cos ( n x ) {\\displaystyle \\cos(nx)}  can be computed from cos ( ( n − 1 ) x ) {\\displaystyle \\cos((n-1)x)} , cos ( ( n − 2 ) x ) {\\displaystyle \\cos((n-2)x)} , and cos ( x ) {\\displaystyle \\cos(x)}  with
cos ( n x ) \= 2 cos x cos ( ( n − 1 ) x ) − cos ( ( n − 2 ) x ) . {\\displaystyle \\cos(nx)=2\\cos x\\cos {\\bigl (}(n-1)x{\\bigr )}-\\cos {\\bigl (}(n-2)x{\\bigr )}.} 
This can be proved by adding together the formulae
cos ( ( n − 1 ) x \+ x ) \= cos ( ( n − 1 ) x ) cos x − sin ( ( n − 1 ) x ) sin x cos ( ( n − 1 ) x − x ) \= cos ( ( n − 1 ) x ) cos x \+ sin ( ( n − 1 ) x ) sin x {\\displaystyle {\\begin{aligned}\\cos {\\bigl (}(n-1)x+x{\\bigr )}&=\\cos {\\bigl (}(n-1)x{\\bigr )}\\cos x-\\sin {\\bigl (}(n-1)x{\\bigr )}\\sin x\\\\\\cos {\\bigl (}(n-1)x-x{\\bigr )}&=\\cos {\\bigl (}(n-1)x{\\bigr )}\\cos x+\\sin {\\bigl (}(n-1)x{\\bigr )}\\sin x\\end{aligned}}} 
It follows by induction that cos ( n x ) {\\displaystyle \\cos(nx)}  is a polynomial of cos x , {\\displaystyle \\cos x,}  the so-called Chebyshev polynomial of the first kind, see [Chebyshev polynomials\#Trigonometric definition](https://en.wikipedia.org/wiki/Chebyshev_polynomials#Trigonometric_definition "Chebyshev polynomials").
Similarly, sin ( n x ) {\\displaystyle \\sin(nx)}  can be computed from sin ( ( n − 1 ) x ) , {\\displaystyle \\sin((n-1)x),}  sin ( ( n − 2 ) x ) , {\\displaystyle \\sin((n-2)x),}  and cos x {\\displaystyle \\cos x}  with sin ( n x ) \= 2 cos x sin ( ( n − 1 ) x ) − sin ( ( n − 2 ) x ) {\\displaystyle \\sin(nx)=2\\cos x\\sin {\\bigl (}(n-1)x{\\bigr )}-\\sin {\\bigl (}(n-2)x{\\bigr )}} 
This can be proved by adding formulae for sin ( ( n − 1 ) x \+ x ) {\\displaystyle \\sin((n-1)x+x)}  and sin ( ( n − 1 ) x − x ) . {\\displaystyle \\sin((n-1)x-x).} 
Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:
tan ( n x ) \= tan ( ( n − 1 ) x ) \+ tan x 1 − tan ( ( n − 1 ) x ) tan x . {\\displaystyle \\tan(nx)={\\frac {\\tan {\\bigl (}(n-1)x{\\bigr )}+\\tan x}{1-\\tan {\\bigl (}(n-1)x{\\bigr )}\\tan x}}\\,.} 
### Half-angle formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=18 "Edit section: Half-angle formulas")\]
sin θ 2 \= sgn ( sin θ 2 ) 1 − cos θ 2 cos θ 2 \= sgn ( cos θ 2 ) 1 \+ cos θ 2 tan θ 2 \= 1 − cos θ sin θ \= sin θ 1 \+ cos θ \= csc θ − cot θ \= tan θ 1 \+ sec θ \= sgn ( sin θ ) 1 − cos θ 1 \+ cos θ \= − 1 \+ sgn ( cos θ ) 1 \+ tan 2 θ tan θ cot θ 2 \= 1 \+ cos θ sin θ \= sin θ 1 − cos θ \= csc θ \+ cot θ \= sgn ( sin θ ) 1 \+ cos θ 1 − cos θ sec θ 2 \= sgn ( cos θ 2 ) 2 1 \+ cos θ csc θ 2 \= sgn ( sin θ 2 ) 2 1 − cos θ {\\displaystyle {\\begin{aligned}\\sin {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\sin {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {1-\\cos \\theta }{2}}}\\\\\[3pt\]\\cos {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\cos {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {1+\\cos \\theta }{2}}}\\\\\[3pt\]\\tan {\\frac {\\theta }{2}}&={\\frac {1-\\cos \\theta }{\\sin \\theta }}={\\frac {\\sin \\theta }{1+\\cos \\theta }}=\\csc \\theta -\\cot \\theta ={\\frac {\\tan \\theta }{1+\\sec {\\theta }}}\\\\\[6mu\]&=\\operatorname {sgn}(\\sin \\theta ){\\sqrt {\\frac {1-\\cos \\theta }{1+\\cos \\theta }}}={\\frac {-1+\\operatorname {sgn}(\\cos \\theta ){\\sqrt {1+\\tan ^{2}\\theta }}}{\\tan \\theta }}\\\\\[3pt\]\\cot {\\frac {\\theta }{2}}&={\\frac {1+\\cos \\theta }{\\sin \\theta }}={\\frac {\\sin \\theta }{1-\\cos \\theta }}=\\csc \\theta +\\cot \\theta =\\operatorname {sgn}(\\sin \\theta ){\\sqrt {\\frac {1+\\cos \\theta }{1-\\cos \\theta }}}\\\\\\sec {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\cos {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {2}{1+\\cos \\theta }}}\\\\\\csc {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\sin {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {2}{1-\\cos \\theta }}}\\\\\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\sin {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\sin {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {1-\\cos \\theta }{2}}}\\\\\[3pt\]\\cos {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\cos {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {1+\\cos \\theta }{2}}}\\\\\[3pt\]\\tan {\\frac {\\theta }{2}}&={\\frac {1-\\cos \\theta }{\\sin \\theta }}={\\frac {\\sin \\theta }{1+\\cos \\theta }}=\\csc \\theta -\\cot \\theta ={\\frac {\\tan \\theta }{1+\\sec {\\theta }}}\\\\\[6mu\]&=\\operatorname {sgn} (\\sin \\theta ){\\sqrt {\\frac {1-\\cos \\theta }{1+\\cos \\theta }}}={\\frac {-1+\\operatorname {sgn} (\\cos \\theta ){\\sqrt {1+\\tan ^{2}\\theta }}}{\\tan \\theta }}\\\\\[3pt\]\\cot {\\frac {\\theta }{2}}&={\\frac {1+\\cos \\theta }{\\sin \\theta }}={\\frac {\\sin \\theta }{1-\\cos \\theta }}=\\csc \\theta +\\cot \\theta =\\operatorname {sgn} (\\sin \\theta ){\\sqrt {\\frac {1+\\cos \\theta }{1-\\cos \\theta }}}\\\\\\sec {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\cos {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {2}{1+\\cos \\theta }}}\\\\\\csc {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\sin {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {2}{1-\\cos \\theta }}}\\\\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d53d104c73c4b28f1dd7bfa5d2b0c7207c68cd87) [\[26\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-ReferenceA-26)[\[27\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_half_angle-27)
Also tan η ± θ 2 \= sin η ± sin θ cos η \+ cos θ tan ( θ 2 \+ π 4 ) \= sec θ \+ tan θ 1 − sin θ 1 \+ sin θ \= \| 1 − tan θ 2 \| \| 1 \+ tan θ 2 \| {\\displaystyle {\\begin{aligned}\\tan {\\frac {\\eta \\pm \\theta }{2}}&={\\frac {\\sin \\eta \\pm \\sin \\theta }{\\cos \\eta +\\cos \\theta }}\\\\\[3pt\]\\tan \\left({\\frac {\\theta }{2}}+{\\frac {\\pi }{4}}\\right)&=\\sec \\theta +\\tan \\theta \\\\\[3pt\]{\\sqrt {\\frac {1-\\sin \\theta }{1+\\sin \\theta }}}&={\\frac {\\left\|1-\\tan {\\frac {\\theta }{2}}\\right\|}{\\left\|1+\\tan {\\frac {\\theta }{2}}\\right\|}}\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\tan {\\frac {\\eta \\pm \\theta }{2}}&={\\frac {\\sin \\eta \\pm \\sin \\theta }{\\cos \\eta +\\cos \\theta }}\\\\\[3pt\]\\tan \\left({\\frac {\\theta }{2}}+{\\frac {\\pi }{4}}\\right)&=\\sec \\theta +\\tan \\theta \\\\\[3pt\]{\\sqrt {\\frac {1-\\sin \\theta }{1+\\sin \\theta }}}&={\\frac {\\left\|1-\\tan {\\frac {\\theta }{2}}\\right\|}{\\left\|1+\\tan {\\frac {\\theta }{2}}\\right\|}}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/639270245e163f1afa8682ac8aa12086db43dce1)
### Table
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=19 "Edit section: Table")\]
See also: [Tangent half-angle formula](https://en.wikipedia.org/wiki/Tangent_half-angle_formula "Tangent half-angle formula")
These can be shown by using either the sum and difference identities or the multiple-angle formulae.
| | Sine | Cosine | Tangent | Cotangent |
|---|---|---|---|---|
| Double-angle formula[\[28\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-28)[\[29\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_double_angle-29) | sin ( 2 θ ) \= 2 sin θ cos θ \= 2 tan θ 1 \+ tan 2 θ {\\displaystyle {\\begin{aligned}\\sin(2\\theta )&=2\\sin \\theta \\cos \\theta \\ \\\\&={\\frac {2\\tan \\theta }{1+\\tan ^{2}\\theta }}\\end{aligned}}}  | | | |
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a [compass and straightedge construction](https://en.wikipedia.org/wiki/Compass_and_straightedge_constructions "Compass and straightedge constructions") of [angle trisection](https://en.wikipedia.org/wiki/Angle_trisection "Angle trisection") to the algebraic problem of solving a [cubic equation](https://en.wikipedia.org/wiki/Cubic_function "Cubic function"), which allows one to prove that [trisection is in general impossible](https://en.wikipedia.org/wiki/Angle_trisection#Proof_of_impossibility "Angle trisection") using the given tools.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the [cubic equation](https://en.wikipedia.org/wiki/Cubic_function "Cubic function") 4*x*3 − 3*x* + *d* = 0, where x is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the [discriminant](https://en.wikipedia.org/wiki/Discriminant "Discriminant") of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). [None of these solutions are reducible](https://en.wikipedia.org/wiki/Casus_irreducibilis "Casus irreducibilis") to a real [algebraic expression](https://en.wikipedia.org/wiki/Algebraic_expression "Algebraic expression"), as they use intermediate complex numbers under the [cube roots](https://en.wikipedia.org/wiki/Cube_root "Cube root").
## Power-reduction formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=20 "Edit section: Power-reduction formulae")\]
Obtained by solving the second and third versions of the cosine double-angle formula.
| Sine | Cosine | Other |
|---|---|---|
| sin 2 θ \= 1 − cos ( 2 θ ) 2 {\\displaystyle \\sin ^{2}\\theta ={\\frac {1-\\cos(2\\theta )}{2}}}  | | |
[](https://en.wikipedia.org/wiki/File:Diagram_showing_how_to_derive_the_power_reduction_formula_for_cosine.svg)
Cosine power-reduction formula: an illustrative diagram. The red, orange and blue triangles are all similar, and the red and orange triangles are congruent. The hypotenuse
A
D
¯
{\\displaystyle {\\overline {AD}}}
of the blue triangle has length
2
cos
θ
{\\displaystyle 2\\cos \\theta }
. The angle
∠
D
A
E
{\\displaystyle \\angle DAE}
is
θ
{\\displaystyle \\theta }
, so the base
A
E
¯
{\\displaystyle {\\overline {AE}}}
of that triangle has length
2
cos
2
θ
{\\displaystyle 2\\cos ^{2}\\theta }
. That length is also equal to the summed lengths of
B
D
¯
{\\displaystyle {\\overline {BD}}}
and
A
F
¯
{\\displaystyle {\\overline {AF}}}
, i.e.
1
\+
cos
(
2
θ
)
{\\displaystyle 1+\\cos(2\\theta )}
. Therefore,
2
cos
2
θ
\=
1
\+
cos
(
2
θ
)
{\\displaystyle 2\\cos ^{2}\\theta =1+\\cos(2\\theta )}
. Dividing both sides by
2
{\\displaystyle 2}
yields the power-reduction formula for cosine:
cos
2
θ
\=
{\\displaystyle \\cos ^{2}\\theta =}
1
2
(
1
\+
cos
(
2
θ
)
)
{\\textstyle {\\frac {1}{2}}(1+\\cos(2\\theta ))}
. The half-angle formula for cosine can be obtained by replacing
θ
{\\displaystyle \\theta }
with
θ
/
2
{\\displaystyle \\theta /2}
and taking the square-root of both sides:
cos
(
θ
/
2
)
\=
±
(
1
\+
cos
θ
)
/
2
.
{\\textstyle \\cos \\left(\\theta /2\\right)=\\pm {\\sqrt {\\left(1+\\cos \\theta \\right)/2}}.}
[](https://en.wikipedia.org/wiki/File:Diagram_showing_how_to_derive_the_power_reducing_formula_for_sine.svg)
Sine power-reduction formula: an illustrative diagram. The shaded blue and green triangles, and the red-outlined triangle
E
B
D
{\\displaystyle EBD}
are all right-angled and similar, and all contain the angle
θ
{\\displaystyle \\theta }
. The hypotenuse
B
D
¯
{\\displaystyle {\\overline {BD}}}
of the red-outlined triangle has length
2
sin
θ
{\\displaystyle 2\\sin \\theta }
, so its side
D
E
¯
{\\displaystyle {\\overline {DE}}}
has length
2
sin
2
θ
{\\displaystyle 2\\sin ^{2}\\theta }
. The line segment
A
E
¯
{\\displaystyle {\\overline {AE}}}
has length
cos
2
θ
{\\displaystyle \\cos 2\\theta }
and sum of the lengths of
A
E
¯
{\\displaystyle {\\overline {AE}}}
and
D
E
¯
{\\displaystyle {\\overline {DE}}}
equals the length of
A
D
¯
{\\displaystyle {\\overline {AD}}}
, which is 1. Therefore,
cos
2
θ
\+
2
sin
2
θ
\=
1
{\\displaystyle \\cos 2\\theta +2\\sin ^{2}\\theta =1}
. Subtracting
cos
2
θ
{\\displaystyle \\cos 2\\theta }
from both sides and dividing by 2 by two yields the power-reduction formula for sine:
sin
2
θ
\=
{\\displaystyle \\sin ^{2}\\theta =}
1
2
(
1
−
cos
(
2
θ
)
)
{\\textstyle {\\frac {1}{2}}(1-\\cos(2\\theta ))}
. The half-angle formula for sine can be obtained by replacing
θ
{\\displaystyle \\theta }
with
θ
/
2
{\\displaystyle \\theta /2}
and taking the square-root of both sides:
sin
(
θ
/
2
)
\=
±
(
1
−
cos
θ
)
/
2
.
{\\textstyle \\sin \\left(\\theta /2\\right)=\\pm {\\sqrt {\\left(1-\\cos \\theta \\right)/2}}.}
Note that this figure also illustrates, in the vertical line segment
E
B
¯
{\\displaystyle {\\overline {EB}}}
, that
sin
2
θ
\=
2
sin
θ
cos
θ
{\\displaystyle \\sin 2\\theta =2\\sin \\theta \\cos \\theta }
.
In general terms of powers of sin θ {\\displaystyle \\sin \\theta }  or cos θ {\\displaystyle \\cos \\theta }  the following is true, and can be deduced using [De Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula "De Moivre's formula"), [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") and the [binomial theorem](https://en.wikipedia.org/wiki/Binomial_theorem "Binomial theorem").
| if *n* is ... | cos n θ {\\displaystyle \\cos ^{n}\\theta }  |
|---|---|
## Product-to-sum and sum-to-product identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=21 "Edit section: Product-to-sum and sum-to-product identities")\]
[](https://en.wikipedia.org/wiki/File:Visual_proof_prosthaphaeresis_cosine_formula.svg)
Proof of the sum-and-difference-to-product cosine identity for prosthaphaeresis calculations using an [isosceles triangle](https://en.wikipedia.org/wiki/Isosceles_triangle "Isosceles triangle")
The product-to-sum identities[\[31\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-31) or [prosthaphaeresis](https://en.wikipedia.org/wiki/Prosthaphaeresis "Prosthaphaeresis") formulae can be proven by expanding their right-hand sides using the [angle addition theorems](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities). Historically, the first four of these were known as **Werner's formulas**, after [Johannes Werner](https://en.wikipedia.org/wiki/Johannes_Werner "Johannes Werner") who used them for astronomical calculations.[\[32\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-32) See [amplitude modulation](https://en.wikipedia.org/wiki/Amplitude_modulation#Simplified_analysis_of_standard_AM "Amplitude modulation") for an application of the product-to-sum formulae, and [beat (acoustics)](https://en.wikipedia.org/wiki/Beat_\(acoustics\) "Beat (acoustics)") and [phase detector](https://en.wikipedia.org/wiki/Phase_detector "Phase detector") for applications of the sum-to-product formulae.
### Product-to-sum identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=22 "Edit section: Product-to-sum identities")\]
The product of two sines or cosines of different angles can be converted to a sum of trigonometric functions of a sum and difference of those angles:
cos θ cos φ \= 1 2 ( cos ( θ − φ ) \+ cos ( θ \+ φ ) ) , sin θ sin φ \= 1 2 ( cos ( θ − φ ) − cos ( θ \+ φ ) ) , sin θ cos φ \= 1 2 ( sin ( θ \+ φ ) \+ sin ( θ − φ ) ) , cos θ sin φ \= 1 2 ( sin ( θ \+ φ ) − sin ( θ − φ ) ) . {\\displaystyle {\\begin{aligned}\\cos \\theta \\,\\cos \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cos(\\theta -\\varphi )+\\cos(\\theta +\\varphi ){\\bigr )},\\\\\[5mu\]\\sin \\theta \\,\\sin \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cos(\\theta -\\varphi )-\\cos(\\theta +\\varphi ){\\bigr )},\\\\\[5mu\]\\sin \\theta \\,\\cos \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sin(\\theta +\\varphi )+\\sin(\\theta -\\varphi ){\\bigr )},\\\\\[5mu\]\\cos \\theta \\,\\sin \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sin(\\theta +\\varphi )-\\sin(\\theta -\\varphi ){\\bigr )}.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\cos \\theta \\,\\cos \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cos(\\theta -\\varphi )+\\cos(\\theta +\\varphi ){\\bigr )},\\\\\[5mu\]\\sin \\theta \\,\\sin \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cos(\\theta -\\varphi )-\\cos(\\theta +\\varphi ){\\bigr )},\\\\\[5mu\]\\sin \\theta \\,\\cos \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sin(\\theta +\\varphi )+\\sin(\\theta -\\varphi ){\\bigr )},\\\\\[5mu\]\\cos \\theta \\,\\sin \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sin(\\theta +\\varphi )-\\sin(\\theta -\\varphi ){\\bigr )}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd715f4b6380798a701537d9c067f28cc371efa) As a corollary, the product or quotient of tangents can be converted to a quotient of sums of cosines or sines, respectively, tan θ tan φ \= cos ( θ − φ ) − cos ( θ \+ φ ) cos ( θ − φ ) \+ cos ( θ \+ φ ) , tan θ tan φ \= sin ( θ \+ φ ) \+ sin ( θ − φ ) sin ( θ \+ φ ) − sin ( θ − φ ) . {\\displaystyle {\\begin{aligned}\\tan \\theta \\,\\tan \\varphi &={\\frac {\\cos(\\theta -\\varphi )-\\cos(\\theta +\\varphi )}{\\cos(\\theta -\\varphi )+\\cos(\\theta +\\varphi )}},\\\\\[5mu\]{\\frac {\\tan \\theta }{\\tan \\varphi }}&={\\frac {\\sin(\\theta +\\varphi )+\\sin(\\theta -\\varphi )}{\\sin(\\theta +\\varphi )-\\sin(\\theta -\\varphi )}}.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\tan \\theta \\,\\tan \\varphi &={\\frac {\\cos(\\theta -\\varphi )-\\cos(\\theta +\\varphi )}{\\cos(\\theta -\\varphi )+\\cos(\\theta +\\varphi )}},\\\\\[5mu\]{\\frac {\\tan \\theta }{\\tan \\varphi }}&={\\frac {\\sin(\\theta +\\varphi )+\\sin(\\theta -\\varphi )}{\\sin(\\theta +\\varphi )-\\sin(\\theta -\\varphi )}}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69a3a5dee2394b1ed85551729df20f0abd6770b7)
More generally, for a product of any number of sines or cosines,\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] ∏ k \= 1 n cos θ k \= 1 2 n ∑ e ∈ S cos ( e 1 θ 1 \+ ⋯ \+ e n θ n ) where e \= ( e 1 , … , e n ) ∈ S \= { 1 , − 1 } n , ∏ k \= 1 n sin θ k \= ( − 1 ) ⌊ n 2 ⌋ 2 n { ∑ e ∈ S cos ( e 1 θ 1 \+ ⋯ \+ e n θ n ) ∏ j \= 1 n e j if n is even , ∑ e ∈ S sin ( e 1 θ 1 \+ ⋯ \+ e n θ n ) ∏ j \= 1 n e j if n is odd . {\\displaystyle {\\begin{aligned}\\prod \_{k=1}^{n}\\cos \\theta \_{k}&={\\frac {1}{2^{n}}}\\sum \_{e\\in S}\\cos(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\\\\[5mu\]&{\\text{where }}e=(e\_{1},\\ldots ,e\_{n})\\in S=\\{1,-1\\}^{n},\\\\\\prod \_{k=1}^{n}\\sin \\theta \_{k}&={\\frac {(-1)^{\\left\\lfloor {\\frac {n}{2}}\\right\\rfloor }}{2^{n}}}{\\begin{cases}\\displaystyle \\sum \_{e\\in S}\\cos(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\prod \_{j=1}^{n}e\_{j}\\;{\\text{if}}\\;n\\;{\\text{is even}},\\\\\\displaystyle \\sum \_{e\\in S}\\sin(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\prod \_{j=1}^{n}e\_{j}\\;{\\text{if}}\\;n\\;{\\text{is odd}}.\\end{cases}}\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\prod \_{k=1}^{n}\\cos \\theta \_{k}&={\\frac {1}{2^{n}}}\\sum \_{e\\in S}\\cos(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\\\\[5mu\]&{\\text{where }}e=(e\_{1},\\ldots ,e\_{n})\\in S=\\{1,-1\\}^{n},\\\\\\prod \_{k=1}^{n}\\sin \\theta \_{k}&={\\frac {(-1)^{\\left\\lfloor {\\frac {n}{2}}\\right\\rfloor }}{2^{n}}}{\\begin{cases}\\displaystyle \\sum \_{e\\in S}\\cos(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\prod \_{j=1}^{n}e\_{j}\\;{\\text{if}}\\;n\\;{\\text{is even}},\\\\\\displaystyle \\sum \_{e\\in S}\\sin(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\prod \_{j=1}^{n}e\_{j}\\;{\\text{if}}\\;n\\;{\\text{is odd}}.\\end{cases}}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/106ce6115e77f036a95a2096bc66b01927e5dff9)
### Sum-to-product identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=23 "Edit section: Sum-to-product identities")\]
[](https://en.wikipedia.org/wiki/File:Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg)
Diagram illustrating sum-to-product identities for sine and cosine. The blue right-angled triangle has angle
θ
{\\displaystyle \\theta }
and the red right-angled triangle has angle
φ
{\\displaystyle \\varphi }
. Both have a hypotenuse of length 1. Auxiliary angles, here called
p
{\\displaystyle p}
and
q
{\\displaystyle q}
, are constructed such that
p
\=
1
2
(
θ
\+
φ
)
{\\displaystyle p={\\tfrac {1}{2}}(\\theta +\\varphi )}
and
q
\=
1
2
(
θ
−
φ
)
{\\displaystyle q={\\tfrac {1}{2}}(\\theta -\\varphi )}
. Therefore,
θ
\=
p
\+
q
{\\displaystyle \\theta =p+q}
and
φ
\=
p
−
q
{\\displaystyle \\varphi =p-q}
. This allows the two congruent purple-outline triangles
A
F
G
{\\displaystyle AFG}
and
F
C
E
{\\displaystyle FCE}
to be constructed, each with hypotenuse
cos
q
{\\displaystyle \\cos q}
and angle
p
{\\displaystyle p}
at their base. The sum of the heights of the red and blue triangles is
sin
θ
\+
sin
φ
{\\displaystyle \\sin \\theta +\\sin \\varphi }
, and this is equal to twice the height of one purple triangle, i.e.
2
sin
p
cos
q
{\\displaystyle 2\\sin p\\cos q}
. Writing
p
{\\displaystyle p}
and
q
{\\displaystyle q}
in that equation in terms of
θ
{\\displaystyle \\theta }
and
φ
{\\displaystyle \\varphi }
yields a sum-to-product identity for sine:
sin
θ
\+
sin
φ
\=
2
sin
1
2
(
θ
\+
φ
)
cos
1
2
(
θ
−
φ
)
{\\displaystyle \\sin \\theta +\\sin \\varphi =2\\sin {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\cos {\\tfrac {1}{2}}(\\theta -\\varphi )}
. Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.
The sum of sines or cosines of two angles can be converted to a product of sines or cosines of the mean and half the difference of the angles:[\[33\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-A&S_sum-to-product-33)
sin θ \+ sin φ \= 2 sin 1 2 ( θ \+ φ ) cos 1 2 ( θ − φ ) , sin θ − sin φ \= 2 cos 1 2 ( θ \+ φ ) sin 1 2 ( θ − φ ) , cos θ \+ cos φ \= 2 cos 1 2 ( θ \+ φ ) cos 1 2 ( θ − φ ) , cos θ − cos φ \= − 2 sin 1 2 ( θ \+ φ ) sin 1 2 ( θ − φ ) . {\\displaystyle {\\begin{aligned}\\sin \\theta +\\sin \\varphi &=2\\sin {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\cos {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\sin \\theta -\\sin \\varphi &=2\\cos {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\sin {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\cos \\theta +\\cos \\varphi &=2\\cos {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\cos {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\cos \\theta -\\cos \\varphi &=-2\\sin {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\sin {\\tfrac {1}{2}}(\\theta -\\varphi ).\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\sin \\theta +\\sin \\varphi &=2\\sin {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\cos {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\sin \\theta -\\sin \\varphi &=2\\cos {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\sin {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\cos \\theta +\\cos \\varphi &=2\\cos {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\cos {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\cos \\theta -\\cos \\varphi &=-2\\sin {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\sin {\\tfrac {1}{2}}(\\theta -\\varphi ).\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3676cbeab42be735d5fb198d843eeec23ee9b763)
The sum of the tangent of two angles can be converted to a quotient of the sine of angles divided by the product of the cosines:[\[33\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-A&S_sum-to-product-33) tan θ ± tan φ \= sin ( θ ± φ ) cos θ cos φ . {\\displaystyle \\tan \\theta \\pm \\tan \\varphi ={\\frac {\\sin(\\theta \\pm \\varphi )}{\\cos \\theta \\,\\cos \\varphi }}.} 
### Hermite's cotangent identity
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=24 "Edit section: Hermite's cotangent identity")\]
Main article: [Hermite's cotangent identity](https://en.wikipedia.org/wiki/Hermite%27s_cotangent_identity "Hermite's cotangent identity")
[Charles Hermite](https://en.wikipedia.org/wiki/Charles_Hermite "Charles Hermite") demonstrated the following identity.[\[34\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-34) Suppose a 1 , … , a n {\\displaystyle a\_{1},\\ldots ,a\_{n}}  are [complex numbers](https://en.wikipedia.org/wiki/Complex_number "Complex number"), no two of which differ by an integer multiple of π. Let
A n , k \= ∏ 1 ≤ j ≤ n j ≠ k cot ( a k − a j ) {\\displaystyle A\_{n,k}=\\prod \_{\\begin{smallmatrix}1\\leq j\\leq n\\\\j\\neq k\\end{smallmatrix}}\\cot(a\_{k}-a\_{j})} 
(in particular, A 1 , 1 , {\\displaystyle A\_{1,1},}  being an [empty product](https://en.wikipedia.org/wiki/Empty_product "Empty product"), is 1). Then
cot ( z − a 1 ) ⋯ cot ( z − a n ) \= cos n π 2 \+ ∑ k \= 1 n A n , k cot ( z − a k ) . {\\displaystyle \\cot(z-a\_{1})\\cdots \\cot(z-a\_{n})=\\cos {\\frac {n\\pi }{2}}+\\sum \_{k=1}^{n}A\_{n,k}\\cot(z-a\_{k}).} 
The simplest non-trivial example is the case *n* = 2:
cot ( z − a 1 ) cot ( z − a 2 ) \= − 1 \+ cot ( a 1 − a 2 ) cot ( z − a 1 ) \+ cot ( a 2 − a 1 ) cot ( z − a 2 ) . {\\displaystyle \\cot(z-a\_{1})\\cot(z-a\_{2})=-1+\\cot(a\_{1}-a\_{2})\\cot(z-a\_{1})+\\cot(a\_{2}-a\_{1})\\cot(z-a\_{2}).} 
### Finite products of trigonometric functions
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=25 "Edit section: Finite products of trigonometric functions")\]
For [coprime](https://en.wikipedia.org/wiki/Coprime "Coprime") integers n, m
∏ k \= 1 n ( 2 a \+ 2 cos ( 2 π k m n \+ x ) ) \= 2 ( T n ( a ) \+ ( − 1 ) n \+ m cos ( n x ) ) {\\displaystyle \\prod \_{k=1}^{n}\\left(2a+2\\cos \\left({\\frac {2\\pi km}{n}}+x\\right)\\right)=2\\left(T\_{n}(a)+{(-1)}^{n+m}\\cos(nx)\\right)} 
where Tn is the [Chebyshev polynomial](https://en.wikipedia.org/wiki/Chebyshev_polynomial "Chebyshev polynomial").\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
The following relationship holds for the sine function
∏ k \= 1 n − 1 sin ( k π n ) \= n 2 n − 1 . {\\displaystyle \\prod \_{k=1}^{n-1}\\sin \\left({\\frac {k\\pi }{n}}\\right)={\\frac {n}{2^{n-1}}}.} 
More generally for an integer *n* \> 0[\[35\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-35)
sin ( n x ) \= 2 n − 1 ∏ k \= 0 n − 1 sin ( k π n \+ x ) \= 2 n − 1 ∏ k \= 1 n sin ( k π n − x ) . {\\displaystyle \\sin(nx)=2^{n-1}\\prod \_{k=0}^{n-1}\\sin \\left({\\frac {k\\pi }{n}}+x\\right)=2^{n-1}\\prod \_{k=1}^{n}\\sin \\left({\\frac {k\\pi }{n}}-x\\right).} 
or written in terms of the [chord](https://en.wikipedia.org/wiki/Chord_\(geometry\) "Chord (geometry)") function crd x ≡ 2 sin 1 2 x {\\textstyle \\operatorname {crd} x\\equiv 2\\sin {\\tfrac {1}{2}}x} ,
crd ( n x ) \= ∏ k \= 1 n crd ( 2 k π n − x ) . {\\displaystyle \\operatorname {crd} (nx)=\\prod \_{k=1}^{n}\\operatorname {crd} \\left({\\frac {2k\\pi }{n}}-x\\right).} 
This comes from the [factorization of the polynomial](https://en.wikipedia.org/wiki/Factorization_of_polynomials "Factorization of polynomials") z n − 1 {\\textstyle z^{n}-1}  into linear factors (cf. [root of unity](https://en.wikipedia.org/wiki/Root_of_unity "Root of unity")): For any complex z and an integer *n* \> 0,
z n − 1 \= ∏ k \= 1 n ( z − exp 2 k i π n ) . {\\displaystyle z^{n}-1=\\prod \_{k=1}^{n}\\left(z-\\exp {\\frac {2ki\\pi }{n}}\\right).} 
## Linear combinations
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=26 "Edit section: Linear combinations")\]
For some purposes it is important to know that any [linear combination](https://en.wikipedia.org/wiki/Linear_combination "Linear combination") of sine waves of the same period or frequency but different [phase shifts](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)") is also a sine wave with the same period or frequency, but a different phase shift. This is useful in [sinusoid](https://en.wikipedia.org/wiki/Sinusoid "Sinusoid") [data fitting](https://en.wikipedia.org/wiki/Data_fitting "Data fitting"), because the measured or observed data are linearly related to the a and b unknowns of the [in-phase and quadrature components](https://en.wikipedia.org/wiki/In-phase_and_quadrature_components "In-phase and quadrature components") basis below, resulting in a simpler [Jacobian](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant "Jacobian matrix and determinant"), compared to that of c {\\displaystyle c}  and φ {\\displaystyle \\varphi } .
### Sine and cosine
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=27 "Edit section: Sine and cosine")\]
The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,[\[36\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-36)[\[37\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-ReferenceB-37)
a cos x \+ b sin x \= c cos ( x \+ φ ) {\\displaystyle a\\cos x+b\\sin x=c\\cos(x+\\varphi )} 
where c {\\displaystyle c}  and φ {\\displaystyle \\varphi }  are defined as so:
c \= sgn ( a ) a 2 \+ b 2 , φ \= arctan ( − b a ) , {\\displaystyle {\\begin{aligned}c&=\\operatorname {sgn}(a){\\sqrt {a^{2}+b^{2}}},\\\\\\varphi &=\\arctan \\left(-{\\frac {b}{a}}\\right),\\end{aligned}}} 
given that a ≠ 0\. {\\displaystyle a\\neq 0.} 
### Arbitrary phase shift
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=28 "Edit section: Arbitrary phase shift")\]
More generally, for arbitrary phase shifts, we have
a sin ( x \+ θ a ) \+ b sin ( x \+ θ b ) \= c sin ( x \+ φ ) {\\displaystyle a\\sin(x+\\theta \_{a})+b\\sin(x+\\theta \_{b})=c\\sin(x+\\varphi )} 
where c {\\displaystyle c}  and φ {\\displaystyle \\varphi }  satisfy:
c 2 \= a 2 \+ b 2 \+ 2 a b cos ( θ a − θ b ) , tan φ \= a sin θ a \+ b sin θ b a cos θ a \+ b cos θ b . {\\displaystyle {\\begin{aligned}c^{2}&=a^{2}+b^{2}+2ab\\cos \\left(\\theta \_{a}-\\theta \_{b}\\right),\\\\\\tan \\varphi &={\\frac {a\\sin \\theta \_{a}+b\\sin \\theta \_{b}}{a\\cos \\theta \_{a}+b\\cos \\theta \_{b}}}.\\end{aligned}}} 
### More than two sinusoids
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=29 "Edit section: More than two sinusoids")\]
See also: [Phasor addition](https://en.wikipedia.org/wiki/Phasor_\(sine_waves\)#Addition "Phasor (sine waves)")
The general case reads[\[37\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-ReferenceB-37)
∑ i a i sin ( x \+ θ i ) \= a sin ( x \+ θ ) , {\\displaystyle \\sum \_{i}a\_{i}\\sin(x+\\theta \_{i})=a\\sin(x+\\theta ),}  where a 2 \= ∑ i , j a i a j cos ( θ i − θ j ) {\\displaystyle a^{2}=\\sum \_{i,j}a\_{i}a\_{j}\\cos(\\theta \_{i}-\\theta \_{j})}  and tan θ \= ∑ i a i sin θ i ∑ i a i cos θ i . {\\displaystyle \\tan \\theta ={\\frac {\\sum \_{i}a\_{i}\\sin \\theta \_{i}}{\\sum \_{i}a\_{i}\\cos \\theta \_{i}}}.} 
## Lagrange's trigonometric identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=30 "Edit section: Lagrange's trigonometric identities")\]
These identities, named after [Joseph Louis Lagrange](https://en.wikipedia.org/wiki/Joseph_Louis_Lagrange "Joseph Louis Lagrange"), are:[\[38\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Muniz-38)[\[39\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-39)[\[40\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-40) ∑ k \= 0 n sin k θ \= cos 1 2 θ − cos ( ( n \+ 1 2 ) θ ) 2 sin 1 2 θ ∑ k \= 1 n cos k θ \= − sin 1 2 θ \+ sin ( ( n \+ 1 2 ) θ ) 2 sin 1 2 θ {\\displaystyle {\\begin{aligned}\\sum \_{k=0}^{n}\\sin k\\theta &={\\frac {\\cos {\\tfrac {1}{2}}\\theta -\\cos \\left(\\left(n+{\\tfrac {1}{2}}\\right)\\theta \\right)}{2\\sin {\\tfrac {1}{2}}\\theta }}\\\\\[5pt\]\\sum \_{k=1}^{n}\\cos k\\theta &={\\frac {-\\sin {\\tfrac {1}{2}}\\theta +\\sin \\left(\\left(n+{\\tfrac {1}{2}}\\right)\\theta \\right)}{2\\sin {\\tfrac {1}{2}}\\theta }}\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\sum \_{k=0}^{n}\\sin k\\theta &={\\frac {\\cos {\\tfrac {1}{2}}\\theta -\\cos \\left(\\left(n+{\\tfrac {1}{2}}\\right)\\theta \\right)}{2\\sin {\\tfrac {1}{2}}\\theta }}\\\\\[5pt\]\\sum \_{k=1}^{n}\\cos k\\theta &={\\frac {-\\sin {\\tfrac {1}{2}}\\theta +\\sin \\left(\\left(n+{\\tfrac {1}{2}}\\right)\\theta \\right)}{2\\sin {\\tfrac {1}{2}}\\theta }}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4679d7bb8112018d06e7d3a849bbe4ce024ff148) for θ ≢ 0 ( mod 2 π ) . {\\displaystyle \\theta \\not \\equiv 0{\\pmod {2\\pi }}.} 
A related function is the [Dirichlet kernel](https://en.wikipedia.org/wiki/Dirichlet_kernel "Dirichlet kernel"):
D n ( θ ) \= 1 \+ 2 ∑ k \= 1 n cos k θ \= sin ( ( n \+ 1 2 ) θ ) sin 1 2 θ . {\\displaystyle D\_{n}(\\theta )=1+2\\sum \_{k=1}^{n}\\cos k\\theta ={\\frac {\\sin \\left(\\left(n+{\\tfrac {1}{2}}\\right)\\theta \\right)}{\\sin {\\tfrac {1}{2}}\\theta }}.} 
A similar identity is[\[41\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-41)
∑ k \= 1 n cos ( 2 k − 1 ) α \= sin ( 2 n α ) 2 sin α . {\\displaystyle \\sum \_{k=1}^{n}\\cos(2k-1)\\alpha ={\\frac {\\sin(2n\\alpha )}{2\\sin \\alpha }}.} 
The proof is the following. By using the [angle sum and difference identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities), sin ( A \+ B ) − sin ( A − B ) \= 2 cos A sin B . {\\displaystyle \\sin(A+B)-\\sin(A-B)=2\\cos A\\sin B.}  Then let's examine the following formula,
2 sin α ∑ k \= 1 n cos ( 2 k − 1 ) α \= 2 sin α cos α \+ 2 sin α cos 3 α \+ 2 sin α cos 5 α \+ ⋯ \+ 2 sin α cos ( 2 n − 1 ) α {\\displaystyle 2\\sin \\alpha \\sum \_{k=1}^{n}\\cos(2k-1)\\alpha =2\\sin \\alpha \\cos \\alpha +2\\sin \\alpha \\cos 3\\alpha +2\\sin \\alpha \\cos 5\\alpha +\\cdots +2\\sin \\alpha \\cos(2n-1)\\alpha }  and this formula can be written by using the above identity,
2 sin α ∑ k \= 1 n cos ( 2 k − 1 ) α \= ∑ k \= 1 n ( sin ( 2 k α ) − sin ( 2 ( k − 1 ) α ) ) \= ( sin 2 α − sin 0 ) \+ ( sin 4 α − sin 2 α ) \+ ( sin 6 α − sin 4 α ) \+ ⋯ \+ ( sin ( 2 n α ) − sin ( 2 ( n − 1 ) α ) ) \= sin ( 2 n α ) . {\\displaystyle {\\begin{aligned}&2\\sin \\alpha \\sum \_{k=1}^{n}\\cos(2k-1)\\alpha \\\\&\\quad =\\sum \_{k=1}^{n}(\\sin(2k\\alpha )-\\sin(2(k-1)\\alpha ))\\\\&\\quad =(\\sin 2\\alpha -\\sin 0)+(\\sin 4\\alpha -\\sin 2\\alpha )+(\\sin 6\\alpha -\\sin 4\\alpha )+\\cdots +(\\sin(2n\\alpha )-\\sin(2(n-1)\\alpha ))\\\\&\\quad =\\sin(2n\\alpha ).\\end{aligned}}} 
So, dividing this formula with 2 sin α {\\displaystyle 2\\sin \\alpha }  completes the proof.
## Certain linear fractional transformations
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=31 "Edit section: Certain linear fractional transformations")\]
If f ( x ) {\\displaystyle f(x)}  is given by the [linear fractional transformation](https://en.wikipedia.org/wiki/M%C3%B6bius_transformation "Möbius transformation") f ( x ) \= ( cos α ) x − sin α ( sin α ) x \+ cos α , {\\displaystyle f(x)={\\frac {(\\cos \\alpha )x-\\sin \\alpha }{(\\sin \\alpha )x+\\cos \\alpha }},}  and similarly g ( x ) \= ( cos β ) x − sin β ( sin β ) x \+ cos β , {\\displaystyle g(x)={\\frac {(\\cos \\beta )x-\\sin \\beta }{(\\sin \\beta )x+\\cos \\beta }},}  then f ( g ( x ) ) \= g ( f ( x ) ) \= ( cos ( α \+ β ) ) x − sin ( α \+ β ) ( sin ( α \+ β ) ) x \+ cos ( α \+ β ) . {\\displaystyle f{\\big (}g(x){\\big )}=g{\\big (}f(x){\\big )}={\\frac {{\\big (}\\cos(\\alpha +\\beta ){\\big )}x-\\sin(\\alpha +\\beta )}{{\\big (}\\sin(\\alpha +\\beta ){\\big )}x+\\cos(\\alpha +\\beta )}}.} 
More tersely stated, if for all α {\\displaystyle \\alpha }  we let f α {\\displaystyle f\_{\\alpha }}  be what we called f {\\displaystyle f}  above, then f α ∘ f β \= f α \+ β . {\\displaystyle f\_{\\alpha }\\circ f\_{\\beta }=f\_{\\alpha +\\beta }.} 
If x {\\displaystyle x}  is the slope of a line, then f ( x ) {\\displaystyle f(x)}  is the slope of its rotation through an angle of − α . {\\displaystyle -\\alpha .} 
## Relation to the complex exponential function
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=32 "Edit section: Relation to the complex exponential function")\]
Main article: [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula")
Euler's formula states that, for any real number *x*:[\[42\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-42) e i x \= cos x \+ i sin x , {\\displaystyle e^{ix}=\\cos x+i\\sin x,}  where *i* is the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit"). Substituting −*x* for *x* gives us: e − i x \= cos ( − x ) \+ i sin ( − x ) \= cos x − i sin x . {\\displaystyle e^{-ix}=\\cos(-x)+i\\sin(-x)=\\cos x-i\\sin x.} 
These two equations can be used to solve for cosine and sine in terms of the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function"). Specifically,[\[43\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-43)[\[44\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-44) cos x \= e i x \+ e − i x 2 {\\displaystyle \\cos x={\\frac {e^{ix}+e^{-ix}}{2}}}  sin x \= e i x − e − i x 2 i {\\displaystyle \\sin x={\\frac {e^{ix}-e^{-ix}}{2i}}} 
These formulae are useful for proving many other trigonometric identities. For example, that *e**i*(*θ*\+*φ*) = *e**iθ* *e**iφ* means that
cos(*θ* + *φ*) + *i* sin(*θ* + *φ*) = (cos *θ* + *i* sin *θ*) (cos *φ* + *i* sin *φ*) = (cos *θ* cos *φ* − sin *θ* sin *φ*) + *i* (cos *θ* sin *φ* + sin *θ* cos *φ*).
That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.
The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the [complex logarithm](https://en.wikipedia.org/wiki/Complex_logarithm "Complex logarithm").
| Function | Inverse function[\[45\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-45) |
|---|---|
| sin θ \= e i θ − e − i θ 2 i {\\displaystyle \\sin \\theta ={\\frac {e^{i\\theta }-e^{-i\\theta }}{2i}}}  | |
## Relation to complex hyperbolic functions
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=33 "Edit section: Relation to complex hyperbolic functions")\]
Trigonometric functions may be deduced from [hyperbolic functions](https://en.wikipedia.org/wiki/Hyperbolic_functions "Hyperbolic functions") with [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") arguments. The formulae for the relations are shown below[\[46\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-46)[\[47\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-47).sin x \= − i sinh ( i x ) cos x \= cosh ( i x ) tan x \= − i tanh ( i x ) cot x \= i coth ( i x ) sec x \= sech ( i x ) csc x \= i csch ( i x ) {\\displaystyle {\\begin{aligned}\\sin x&=-i\\sinh(ix)\\\\\\cos x&=\\cosh(ix)\\\\\\tan x&=-i\\tanh(ix)\\\\\\cot x&=i\\coth(ix)\\\\\\sec x&=\\operatorname {sech} (ix)\\\\\\csc x&=i\\operatorname {csch} (ix)\\\\\\end{aligned}}} 
## Series expansion
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=34 "Edit section: Series expansion")\]
When using a [power series](https://en.wikipedia.org/wiki/Power_series "Power series") expansion to define trigonometric functions, the following identities are obtained:[\[48\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-48)
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{\\displaystyle {\\begin{aligned}\\sin x&=x-{\\frac {x^{3}}{3!}}+{\\frac {x^{5}}{5!}}-{\\frac {x^{7}}{7!}}+\\cdots &&=\\sum \_{n=0}^{\\infty }(-1)^{n}{\\frac {x^{2n+1}}{(2n+1)!}}\\\\\\cos x&=1-{\\frac {x^{2}}{2!}}+{\\frac {x^{4}}{4!}}-{\\frac {x^{6}}{6!}}+\\cdots &&=\\sum \_{n=0}^{\\infty }(-1)^{n}{\\frac {x^{2n}}{(2n)!}}\\end{aligned}}}
## Infinite product formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=35 "Edit section: Infinite product formulae")\]
For applications to [special functions](https://en.wikipedia.org/wiki/Special_functions "Special functions"), the following [infinite product](https://en.wikipedia.org/wiki/Infinite_product "Infinite product") formulae for trigonometric functions are useful:[\[49\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-49)[\[50\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-50)
sin x \= x ∏ n \= 1 ∞ ( 1 − x 2 π 2 n 2 ) , cos x \= ∏ n \= 1 ∞ ( 1 − x 2 π 2 ( n − 1 2 ) ) 2 ) , sinh x \= x ∏ n \= 1 ∞ ( 1 \+ x 2 π 2 n 2 ) , cosh x \= ∏ n \= 1 ∞ ( 1 \+ x 2 π 2 ( n − 1 2 ) ) 2 ) . {\\displaystyle {\\begin{aligned}\\sin x&=x\\prod \_{n=1}^{\\infty }\\left(1-{\\frac {x^{2}}{\\pi ^{2}n^{2}}}\\right),&\\cos x&=\\prod \_{n=1}^{\\infty }\\left(1-{\\frac {x^{2}}{\\pi ^{2}\\left(n-{\\frac {1}{2}}\\right)\\!{\\vphantom {)}}^{2}}}\\right),\\\\\[10mu\]\\sinh x&=x\\prod \_{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{\\pi ^{2}n^{2}}}\\right),&\\cosh x&=\\prod \_{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{\\pi ^{2}\\left(n-{\\frac {1}{2}}\\right)\\!{\\vphantom {)}}^{2}}}\\right).\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\sin x&=x\\prod \_{n=1}^{\\infty }\\left(1-{\\frac {x^{2}}{\\pi ^{2}n^{2}}}\\right),&\\cos x&=\\prod \_{n=1}^{\\infty }\\left(1-{\\frac {x^{2}}{\\pi ^{2}\\left(n-{\\frac {1}{2}}\\right)\\!{\\vphantom {)}}^{2}}}\\right),\\\\\[10mu\]\\sinh x&=x\\prod \_{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{\\pi ^{2}n^{2}}}\\right),&\\cosh x&=\\prod \_{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{\\pi ^{2}\\left(n-{\\frac {1}{2}}\\right)\\!{\\vphantom {)}}^{2}}}\\right).\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d43e1176a38b1e6b653837eb782ae79735783ac)
## Inverse trigonometric functions
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=36 "Edit section: Inverse trigonometric functions")\]
Main article: [Inverse trigonometric functions](https://en.wikipedia.org/wiki/Inverse_trigonometric_functions "Inverse trigonometric functions")
The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[\[51\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-51)
sin ( arcsin x ) \= x cos ( arcsin x ) \= 1 − x 2 tan ( arcsin x ) \= x 1 − x 2 sin ( arccos x ) \= 1 − x 2 cos ( arccos x ) \= x tan ( arccos x ) \= 1 − x 2 x sin ( arctan x ) \= x 1 \+ x 2 cos ( arctan x ) \= 1 1 \+ x 2 tan ( arctan x ) \= x sin ( arccsc x ) \= 1 x cos ( arccsc x ) \= 1 − 1 x 2 tan ( arccsc x ) \= 1 x 1 − 1 x 2 sin ( arcsec x ) \= 1 − 1 x 2 cos ( arcsec x ) \= 1 x tan ( arcsec x ) \= x 1 − 1 x 2 sin ( arccot x ) \= 1 1 \+ x 2 cos ( arccot x ) \= x 1 \+ x 2 tan ( arccot x ) \= 1 x {\\displaystyle {\\begin{aligned}\\sin(\\arcsin x)&=x&\\cos(\\arcsin x)&={\\sqrt {1-x^{2}}}&\\tan(\\arcsin x)&={\\frac {x}{\\sqrt {1-x^{2}}}}\\\\\\sin(\\arccos x)&={\\sqrt {1-x^{2}}}&\\cos(\\arccos x)&=x&\\tan(\\arccos x)&={\\frac {\\sqrt {1-x^{2}}}{x}}\\\\\\sin(\\arctan x)&={\\frac {x}{\\sqrt {1+x^{2}}}}&\\cos(\\arctan x)&={\\frac {1}{\\sqrt {1+x^{2}}}}&\\tan(\\arctan x)&=x\\\\\\sin(\\operatorname {arccsc} x)&={\\frac {1}{x}}&\\cos(\\operatorname {arccsc} x)&={\\sqrt {1-{\\frac {1}{x^{2}}}}}&\\tan(\\operatorname {arccsc} x)&={\\frac {1}{x{\\sqrt {1-{\\frac {1}{x^{2}}}}}}}\\\\\\sin(\\operatorname {arcsec} x)&={\\sqrt {1-{\\frac {1}{x^{2}}}}}&\\cos(\\operatorname {arcsec} x)&={\\frac {1}{x}}&\\tan(\\operatorname {arcsec} x)&=x{\\sqrt {1-{\\frac {1}{x^{2}}}}}\\\\\\sin(\\operatorname {arccot} x)&={\\frac {1}{\\sqrt {1+x^{2}}}}&\\cos(\\operatorname {arccot} x)&={\\frac {x}{\\sqrt {1+x^{2}}}}&\\tan(\\operatorname {arccot} x)&={\\frac {1}{x}}\\\\\\end{aligned}}} 
Taking the [multiplicative inverse](https://en.wikipedia.org/wiki/Multiplicative_inverse "Multiplicative inverse") of both sides of the each equation above results in the equations for csc \= 1 sin , sec \= 1 cos , and cot \= 1 tan . {\\displaystyle \\csc ={\\frac {1}{\\sin }},\\;\\sec ={\\frac {1}{\\cos }},{\\text{ and }}\\cot ={\\frac {1}{\\tan }}.}  The right hand side of the formula above will always be flipped. For example, the equation for cot ( arcsin x ) {\\displaystyle \\cot(\\arcsin x)}  is: cot ( arcsin x ) \= 1 tan ( arcsin x ) \= 1 x 1 − x 2 \= 1 − x 2 x {\\displaystyle \\cot(\\arcsin x)={\\frac {1}{\\tan(\\arcsin x)}}={\\frac {1}{\\frac {x}{\\sqrt {1-x^{2}}}}}={\\frac {\\sqrt {1-x^{2}}}{x}}}  while the equations for csc ( arccos x ) {\\displaystyle \\csc(\\arccos x)}  and sec ( arccos x ) {\\displaystyle \\sec(\\arccos x)}  are: csc ( arccos x ) \= 1 sin ( arccos x ) \= 1 1 − x 2 and sec ( arccos x ) \= 1 cos ( arccos x ) \= 1 x . {\\displaystyle \\csc(\\arccos x)={\\frac {1}{\\sin(\\arccos x)}}={\\frac {1}{\\sqrt {1-x^{2}}}}\\qquad {\\text{ and }}\\quad \\sec(\\arccos x)={\\frac {1}{\\cos(\\arccos x)}}={\\frac {1}{x}}.} 
The following identities are implied by the [reflection identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Reflections). They hold whenever x , r , s , − x , − r , {\\displaystyle x,r,s,-x,-r,}  and − s {\\displaystyle -s}  are in the domains of the relevant functions. π 2 \= arcsin ( x ) \+ arccos ( x ) \= arctan ( r ) \+ arccot ( r ) \= arcsec ( s ) \+ arccsc ( s ) π \= arccos ( x ) \+ arccos ( − x ) \= arccot ( r ) \+ arccot ( − r ) \= arcsec ( s ) \+ arcsec ( − s ) 0 \= arcsin ( x ) \+ arcsin ( − x ) \= arctan ( r ) \+ arctan ( − r ) \= arccsc ( s ) \+ arccsc ( − s ) {\\displaystyle {\\begin{alignedat}{9}{\\frac {\\pi }{2}}~&=~\\arcsin(x)&&+\\arccos(x)~&&=~\\arctan(r)&&+\\operatorname {arccot}(r)~&&=~\\operatorname {arcsec}(s)&&+\\operatorname {arccsc}(s)\\\\\[0.4ex\]\\pi ~&=~\\arccos(x)&&+\\arccos(-x)~&&=~\\operatorname {arccot}(r)&&+\\operatorname {arccot}(-r)~&&=~\\operatorname {arcsec}(s)&&+\\operatorname {arcsec}(-s)\\\\\[0.4ex\]0~&=~\\arcsin(x)&&+\\arcsin(-x)~&&=~\\arctan(r)&&+\\arctan(-r)~&&=~\\operatorname {arccsc}(s)&&+\\operatorname {arccsc}(-s)\\\\\[1.0ex\]\\end{alignedat}}} ![{\\displaystyle {\\begin{alignedat}{9}{\\frac {\\pi }{2}}~&=~\\arcsin(x)&&+\\arccos(x)~&&=~\\arctan(r)&&+\\operatorname {arccot} (r)~&&=~\\operatorname {arcsec} (s)&&+\\operatorname {arccsc} (s)\\\\\[0.4ex\]\\pi ~&=~\\arccos(x)&&+\\arccos(-x)~&&=~\\operatorname {arccot} (r)&&+\\operatorname {arccot} (-r)~&&=~\\operatorname {arcsec} (s)&&+\\operatorname {arcsec} (-s)\\\\\[0.4ex\]0~&=~\\arcsin(x)&&+\\arcsin(-x)~&&=~\\arctan(r)&&+\\arctan(-r)~&&=~\\operatorname {arccsc} (s)&&+\\operatorname {arccsc} (-s)\\\\\[1.0ex\]\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/839db041f421b4af5890106450e2b4d3e9e183d4)
Also,[\[52\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Wu-52) arctan x \+ arctan 1 x \= { π 2 , if x \> 0 − π 2 , if x \< 0 arccot x \+ arccot 1 x \= { π 2 , if x \> 0 3 π 2 , if x \< 0 {\\displaystyle {\\begin{aligned}\\arctan x+\\arctan {\\dfrac {1}{x}}&={\\begin{cases}{\\frac {\\pi }{2}},&{\\text{if }}x\>0\\\\-{\\frac {\\pi }{2}},&{\\text{if }}x\<0\\end{cases}}\\\\\\operatorname {arccot} x+\\operatorname {arccot} {\\dfrac {1}{x}}&={\\begin{cases}{\\frac {\\pi }{2}},&{\\text{if }}x\>0\\\\{\\frac {3\\pi }{2}},&{\\text{if }}x\<0\\end{cases}}\\\\\\end{aligned}}}  arccos 1 x \= arcsec x and arcsec 1 x \= arccos x {\\displaystyle \\arccos {\\frac {1}{x}}=\\operatorname {arcsec} x\\qquad {\\text{ and }}\\qquad \\operatorname {arcsec} {\\frac {1}{x}}=\\arccos x}  arcsin 1 x \= arccsc x and arccsc 1 x \= arcsin x {\\displaystyle \\arcsin {\\frac {1}{x}}=\\operatorname {arccsc} x\\qquad {\\text{ and }}\\qquad \\operatorname {arccsc} {\\frac {1}{x}}=\\arcsin x} 
The [arctangent](https://en.wikipedia.org/wiki/Arctangent "Arctangent") function can be expanded as a series:[\[53\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-53) arctan ( n x ) \= ∑ m \= 1 n arctan x 1 \+ ( m − 1 ) m x 2 {\\displaystyle \\arctan(nx)=\\sum \_{m=1}^{n}\\arctan {\\frac {x}{1+(m-1)mx^{2}}}} 
## Identities without variables
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=37 "Edit section: Identities without variables")\]
In terms of the [arctangent](https://en.wikipedia.org/wiki/Arctangent "Arctangent") function we have[\[52\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Wu-52) arctan 1 2 \= arctan 1 3 \+ arctan 1 7 . {\\displaystyle \\arctan {\\frac {1}{2}}=\\arctan {\\frac {1}{3}}+\\arctan {\\frac {1}{7}}.} 
The curious identity known as [Morrie's law](https://en.wikipedia.org/wiki/Morrie%27s_law "Morrie's law"), cos 20 ∘ ⋅ cos 40 ∘ ⋅ cos 80 ∘ \= 1 8 , {\\displaystyle \\cos 20^{\\circ }\\cdot \\cos 40^{\\circ }\\cdot \\cos 80^{\\circ }={\\frac {1}{8}},} 
is a special case of an identity that contains one variable: ∏ j \= 0 k − 1 cos ( 2 j x ) \= sin ( 2 k x ) 2 k sin x . {\\displaystyle \\prod \_{j=0}^{k-1}\\cos \\left(2^{j}x\\right)={\\frac {\\sin \\left(2^{k}x\\right)}{2^{k}\\sin x}}.} 
Similarly, sin 20 ∘ ⋅ sin 40 ∘ ⋅ sin 80 ∘ \= 3 8 {\\displaystyle \\sin 20^{\\circ }\\cdot \\sin 40^{\\circ }\\cdot \\sin 80^{\\circ }={\\frac {\\sqrt {3}}{8}}}  is a special case of an identity with x \= 20 ∘ {\\displaystyle x=20^{\\circ }} : sin x ⋅ sin ( 60 ∘ − x ) ⋅ sin ( 60 ∘ \+ x ) \= sin 3 x 4 . {\\displaystyle \\sin x\\cdot \\sin \\left(60^{\\circ }-x\\right)\\cdot \\sin \\left(60^{\\circ }+x\\right)={\\frac {\\sin 3x}{4}}.} 
For the case x \= 15 ∘ {\\displaystyle x=15^{\\circ }} , sin 15 ∘ ⋅ sin 45 ∘ ⋅ sin 75 ∘ \= 2 8 , sin 15 ∘ ⋅ sin 75 ∘ \= 1 4 . {\\displaystyle {\\begin{aligned}\\sin 15^{\\circ }\\cdot \\sin 45^{\\circ }\\cdot \\sin 75^{\\circ }&={\\frac {\\sqrt {2}}{8}},\\\\\\sin 15^{\\circ }\\cdot \\sin 75^{\\circ }&={\\frac {1}{4}}.\\end{aligned}}} 
For the case x \= 10 ∘ {\\displaystyle x=10^{\\circ }} , sin 10 ∘ ⋅ sin 50 ∘ ⋅ sin 70 ∘ \= 1 8 . {\\displaystyle \\sin 10^{\\circ }\\cdot \\sin 50^{\\circ }\\cdot \\sin 70^{\\circ }={\\frac {1}{8}}.} 
The same cosine identity is cos x ⋅ cos ( 60 ∘ − x ) ⋅ cos ( 60 ∘ \+ x ) \= cos 3 x 4 . {\\displaystyle \\cos x\\cdot \\cos \\left(60^{\\circ }-x\\right)\\cdot \\cos \\left(60^{\\circ }+x\\right)={\\frac {\\cos 3x}{4}}.} 
Similarly, cos 10 ∘ ⋅ cos 50 ∘ ⋅ cos 70 ∘ \= 3 8 , cos 15 ∘ ⋅ cos 45 ∘ ⋅ cos 75 ∘ \= 2 8 , cos 15 ∘ ⋅ cos 75 ∘ \= 1 4 . {\\displaystyle {\\begin{aligned}\\cos 10^{\\circ }\\cdot \\cos 50^{\\circ }\\cdot \\cos 70^{\\circ }&={\\frac {\\sqrt {3}}{8}},\\\\\\cos 15^{\\circ }\\cdot \\cos 45^{\\circ }\\cdot \\cos 75^{\\circ }&={\\frac {\\sqrt {2}}{8}},\\\\\\cos 15^{\\circ }\\cdot \\cos 75^{\\circ }&={\\frac {1}{4}}.\\end{aligned}}} 
Similarly, tan 50 ∘ ⋅ tan 60 ∘ ⋅ tan 70 ∘ \= tan 80 ∘ , tan 40 ∘ ⋅ tan 30 ∘ ⋅ tan 20 ∘ \= tan 10 ∘ . {\\displaystyle {\\begin{aligned}\\tan 50^{\\circ }\\cdot \\tan 60^{\\circ }\\cdot \\tan 70^{\\circ }&=\\tan 80^{\\circ },\\\\\\tan 40^{\\circ }\\cdot \\tan 30^{\\circ }\\cdot \\tan 20^{\\circ }&=\\tan 10^{\\circ }.\\end{aligned}}} 
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below): cos 24 ∘ \+ cos 48 ∘ \+ cos 96 ∘ \+ cos 168 ∘ \= 1 2 . {\\displaystyle \\cos 24^{\\circ }+\\cos 48^{\\circ }+\\cos 96^{\\circ }+\\cos 168^{\\circ }={\\frac {1}{2}}.} 
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators: cos 2 π 21 \+ cos ( 2 ⋅ 2 π 21 ) \+ cos ( 4 ⋅ 2 π 21 ) \+ cos ( 5 ⋅ 2 π 21 ) \+ cos ( 8 ⋅ 2 π 21 ) \+ cos ( 10 ⋅ 2 π 21 ) \= 1 2 . {\\displaystyle \\cos {\\frac {2\\pi }{21}}+\\cos \\left(2\\cdot {\\frac {2\\pi }{21}}\\right)+\\cos \\left(4\\cdot {\\frac {2\\pi }{21}}\\right)+\\cos \\left(5\\cdot {\\frac {2\\pi }{21}}\\right)+\\cos \\left(8\\cdot {\\frac {2\\pi }{21}}\\right)+\\cos \\left(10\\cdot {\\frac {2\\pi }{21}}\\right)={\\frac {1}{2}}.} 
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are [relatively prime](https://en.wikipedia.org/wiki/Coprime "Coprime") to (or have no [prime factors](https://en.wikipedia.org/wiki/Prime_factor "Prime factor") in common with) 21. The last several examples are corollaries of a basic fact about the irreducible [cyclotomic polynomials](https://en.wikipedia.org/wiki/Cyclotomic_polynomial "Cyclotomic polynomial"): the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the [Möbius function](https://en.wikipedia.org/wiki/M%C3%B6bius_function "Möbius function") evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
Other cosine identities include:[\[54\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-54) 2 cos π 3 \= 1 , 2 cos π 5 × 2 cos 2 π 5 \= 1 , 2 cos π 7 × 2 cos 2 π 7 × 2 cos 3 π 7 \= 1 , {\\displaystyle {\\begin{aligned}2\\cos {\\frac {\\pi }{3}}&=1,\\\\2\\cos {\\frac {\\pi }{5}}\\times 2\\cos {\\frac {2\\pi }{5}}&=1,\\\\2\\cos {\\frac {\\pi }{7}}\\times 2\\cos {\\frac {2\\pi }{7}}\\times 2\\cos {\\frac {3\\pi }{7}}&=1,\\end{aligned}}}  and so forth for all odd numbers, and hence cos π 3 \+ cos π 5 × cos 2 π 5 \+ cos π 7 × cos 2 π 7 × cos 3 π 7 \+ ⋯ \= 1\. {\\displaystyle \\cos {\\frac {\\pi }{3}}+\\cos {\\frac {\\pi }{5}}\\times \\cos {\\frac {2\\pi }{5}}+\\cos {\\frac {\\pi }{7}}\\times \\cos {\\frac {2\\pi }{7}}\\times \\cos {\\frac {3\\pi }{7}}+\\dots =1.} 
Many of those curious identities stem from more general facts like the following:[\[55\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-55) ∏ k \= 1 n − 1 sin k π n \= n 2 n − 1 {\\displaystyle \\prod \_{k=1}^{n-1}\\sin {\\frac {k\\pi }{n}}={\\frac {n}{2^{n-1}}}}  and ∏ k \= 1 n − 1 cos k π n \= sin π n 2 2 n − 1 . {\\displaystyle \\prod \_{k=1}^{n-1}\\cos {\\frac {k\\pi }{n}}={\\frac {\\sin {\\frac {\\pi n}{2}}}{2^{n-1}}}.} 
Combining these gives us ∏ k \= 1 n − 1 tan k π n \= n sin π n 2 {\\displaystyle \\prod \_{k=1}^{n-1}\\tan {\\frac {k\\pi }{n}}={\\frac {n}{\\sin {\\frac {\\pi n}{2}}}}} 
If n is an odd number (n \= 2 m \+ 1 {\\displaystyle n=2m+1} ) we can make use of the symmetries to get ∏ k \= 1 m tan k π 2 m \+ 1 \= 2 m \+ 1 {\\displaystyle \\prod \_{k=1}^{m}\\tan {\\frac {k\\pi }{2m+1}}={\\sqrt {2m+1}}} 
The transfer function of the [Butterworth low pass filter](https://en.wikipedia.org/wiki/Butterworth_filter "Butterworth filter") can be expressed in terms of polynomial and poles. By setting the frequency as the [cutoff frequency](https://en.wikipedia.org/wiki/Cutoff_frequency "Cutoff frequency"), the following identity can be proved: ∏ k \= 1 n sin ( 2 k − 1 ) π 4 n \= ∏ k \= 1 n cos ( 2 k − 1 ) π 4 n \= 2 2 n {\\displaystyle \\prod \_{k=1}^{n}\\sin {\\frac {\\left(2k-1\\right)\\pi }{4n}}=\\prod \_{k=1}^{n}\\cos {\\frac {\\left(2k-1\\right)\\pi }{4n}}={\\frac {\\sqrt {2}}{2^{n}}}} 
### Computing π
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=38 "Edit section: Computing π")\]
An efficient way to [compute π](https://en.wikipedia.org/wiki/Pi "Pi") to a [large number of digits](https://en.wikipedia.org/wiki/Approximations_of_pi "Approximations of pi") is based on the following identity without variables, due to [Machin](https://en.wikipedia.org/wiki/John_Machin "John Machin"). This is known as a [Machin-like formula](https://en.wikipedia.org/wiki/Machin-like_formula "Machin-like formula"): π 4 \= 4 arctan 1 5 − arctan 1 239 {\\displaystyle {\\frac {\\pi }{4}}=4\\arctan {\\frac {1}{5}}-\\arctan {\\frac {1}{239}}}  or, alternatively, by using an identity of [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler"): π 4 \= 5 arctan 1 7 \+ 2 arctan 3 79 {\\displaystyle {\\frac {\\pi }{4}}=5\\arctan {\\frac {1}{7}}+2\\arctan {\\frac {3}{79}}}  or by using [Pythagorean triples](https://en.wikipedia.org/wiki/Pythagorean_triple "Pythagorean triple"): π \= arccos 4 5 \+ arccos 5 13 \+ arccos 16 65 \= arcsin 3 5 \+ arcsin 12 13 \+ arcsin 63 65 . {\\displaystyle \\pi =\\arccos {\\frac {4}{5}}+\\arccos {\\frac {5}{13}}+\\arccos {\\frac {16}{65}}=\\arcsin {\\frac {3}{5}}+\\arcsin {\\frac {12}{13}}+\\arcsin {\\frac {63}{65}}.} 
Others include:[\[56\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Harris-56)[\[52\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Wu-52) π 4 \= arctan 1 2 \+ arctan 1 3 , {\\displaystyle {\\frac {\\pi }{4}}=\\arctan {\\frac {1}{2}}+\\arctan {\\frac {1}{3}},}  π \= arctan 1 \+ arctan 2 \+ arctan 3 , {\\displaystyle \\pi =\\arctan 1+\\arctan 2+\\arctan 3,}  π 4 \= 2 arctan 1 3 \+ arctan 1 7 . {\\displaystyle {\\frac {\\pi }{4}}=2\\arctan {\\frac {1}{3}}+\\arctan {\\frac {1}{7}}.} 
Generally, for numbers *t*1, ..., *t**n*−1 ∈ (−1, 1) for which *θ**n* = Σ*n*−1
*k*\=1 arctan *t**k* ∈ (*π*/4, 3*π*/4), let *t**n* = tan(*π*/2 − *θ**n*) = cot *θ**n*. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are *t*1, ..., *t**n*−1 and its value will be in (−1, 1). In particular, the computed *t**n* will be rational whenever all the *t*1, ..., *t**n*−1 values are rational. With these values, π 2 \= ∑ k \= 1 n arctan ( t k ) π \= ∑ k \= 1 n sgn ( t k ) arccos ( 1 − t k 2 1 \+ t k 2 ) π \= ∑ k \= 1 n arcsin ( 2 t k 1 \+ t k 2 ) π \= ∑ k \= 1 n arctan ( 2 t k 1 − t k 2 ) , {\\displaystyle {\\begin{aligned}{\\frac {\\pi }{2}}&=\\sum \_{k=1}^{n}\\arctan(t\_{k})\\\\\\pi &=\\sum \_{k=1}^{n}\\operatorname {sgn}(t\_{k})\\arccos \\left({\\frac {1-t\_{k}^{2}}{1+t\_{k}^{2}}}\\right)\\\\\\pi &=\\sum \_{k=1}^{n}\\arcsin \\left({\\frac {2t\_{k}}{1+t\_{k}^{2}}}\\right)\\\\\\pi &=\\sum \_{k=1}^{n}\\arctan \\left({\\frac {2t\_{k}}{1-t\_{k}^{2}}}\\right)\\,,\\end{aligned}}} 
where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the *t**k* values is not within (−1, 1). Note that if *t* = *p*/*q* is rational, then the (2*t*, 1 − *t*2, 1 + *t*2) values in the above formulae are proportional to the Pythagorean triple (2*pq*, *q*2 − *p*2, *q*2 + *p*2).
For example, for *n* = 3 terms, π 2 \= arctan ( a b ) \+ arctan ( c d ) \+ arctan ( b d − a c a d \+ b c ) {\\displaystyle {\\frac {\\pi }{2}}=\\arctan \\left({\\frac {a}{b}}\\right)+\\arctan \\left({\\frac {c}{d}}\\right)+\\arctan \\left({\\frac {bd-ac}{ad+bc}}\\right)}  for any *a*, *b*, *c*, *d* \> 0.
### An identity of Euclid
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=39 "Edit section: An identity of Euclid")\]
[Euclid](https://en.wikipedia.org/wiki/Euclid "Euclid") showed in Book XIII, Proposition 10 of his *[Elements](https://en.wikipedia.org/wiki/Euclid%27s_Elements "Euclid's Elements")* that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says: sin 2 18 ∘ \+ sin 2 30 ∘ \= sin 2 36 ∘ . {\\displaystyle \\sin ^{2}18^{\\circ }+\\sin ^{2}30^{\\circ }=\\sin ^{2}36^{\\circ }.} 
[Ptolemy](https://en.wikipedia.org/wiki/Ptolemy "Ptolemy") used this proposition to compute some angles in [his table of chords](https://en.wikipedia.org/wiki/Ptolemy%27s_table_of_chords "Ptolemy's table of chords") in Book I, chapter 11 of *[Almagest](https://en.wikipedia.org/wiki/Almagest "Almagest")*.
## Composition of trigonometric functions
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=40 "Edit section: Composition of trigonometric functions")\]
These identities involve a trigonometric function of a trigonometric function:[\[57\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-57)
cos
(
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sin
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\=
J
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cos
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{\\displaystyle \\cos(t\\sin x)=J\_{0}(t)+2\\sum \_{k=1}^{\\infty }J\_{2k}(t)\\cos(2kx)}
sin
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sin
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\=
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sin
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{\\displaystyle \\sin(t\\sin x)=2\\sum \_{k=0}^{\\infty }J\_{2k+1}(t)\\sin {\\big (}(2k+1)x{\\big )}}
cos
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cos
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{\\displaystyle \\cos(t\\cos x)=J\_{0}(t)+2\\sum \_{k=1}^{\\infty }(-1)^{k}J\_{2k}(t)\\cos(2kx)}
sin
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\=
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cos
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{\\displaystyle \\sin(t\\cos x)=2\\sum \_{k=0}^{\\infty }(-1)^{k}J\_{2k+1}(t)\\cos {\\big (}(2k+1)x{\\big )}}
where Ji are [Bessel functions](https://en.wikipedia.org/wiki/Bessel_function "Bessel function").
## Further "conditional" identities for the case *α* + *β* + *γ* = 180°
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=41 "Edit section: Further \"conditional\" identities for the case α + β + γ = 180°")\]
A **conditional trigonometric identity** is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied.[\[58\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-58) The following formulae apply to arbitrary plane triangles and follow from α \+ β \+ γ \= 180 ∘ , {\\displaystyle \\alpha +\\beta +\\gamma =180^{\\circ },}  as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).[\[59\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-59) tan α \+ tan β \+ tan γ \= tan α tan β tan γ 1 \= cot β cot γ \+ cot γ cot α \+ cot α cot β cot ( α 2 ) \+ cot ( β 2 ) \+ cot ( γ 2 ) \= cot ( α 2 ) cot ( β 2 ) cot ( γ 2 ) 1 \= tan ( β 2 ) tan ( γ 2 ) \+ tan ( γ 2 ) tan ( α 2 ) \+ tan ( α 2 ) tan ( β 2 ) sin α \+ sin β \+ sin γ \= 4 cos ( α 2 ) cos ( β 2 ) cos ( γ 2 ) − sin α \+ sin β \+ sin γ \= 4 cos ( α 2 ) sin ( β 2 ) sin ( γ 2 ) cos α \+ cos β \+ cos γ \= 4 sin ( α 2 ) sin ( β 2 ) sin ( γ 2 ) \+ 1 − cos α \+ cos β \+ cos γ \= 4 sin ( α 2 ) cos ( β 2 ) cos ( γ 2 ) − 1 sin ( 2 α ) \+ sin ( 2 β ) \+ sin ( 2 γ ) \= 4 sin α sin β sin γ − sin ( 2 α ) \+ sin ( 2 β ) \+ sin ( 2 γ ) \= 4 sin α cos β cos γ cos ( 2 α ) \+ cos ( 2 β ) \+ cos ( 2 γ ) \= − 4 cos α cos β cos γ − 1 − cos ( 2 α ) \+ cos ( 2 β ) \+ cos ( 2 γ ) \= − 4 cos α sin β sin γ \+ 1 sin 2 α \+ sin 2 β \+ sin 2 γ \= 2 cos α cos β cos γ \+ 2 − sin 2 α \+ sin 2 β \+ sin 2 γ \= 2 cos α sin β sin γ cos 2 α \+ cos 2 β \+ cos 2 γ \= − 2 cos α cos β cos γ \+ 1 − cos 2 α \+ cos 2 β \+ cos 2 γ \= − 2 cos α sin β sin γ \+ 1 sin 2 ( 2 α ) \+ sin 2 ( 2 β ) \+ sin 2 ( 2 γ ) \= − 2 cos ( 2 α ) cos ( 2 β ) cos ( 2 γ ) \+ 2 cos 2 ( 2 α ) \+ cos 2 ( 2 β ) \+ cos 2 ( 2 γ ) \= 2 cos ( 2 α ) cos ( 2 β ) cos ( 2 γ ) \+ 1 1 \= sin 2 ( α 2 ) \+ sin 2 ( β 2 ) \+ sin 2 ( γ 2 ) \+ 2 sin ( α 2 ) sin ( β 2 ) sin ( γ 2 ) {\\displaystyle {\\begin{aligned}\\tan \\alpha +\\tan \\beta +\\tan \\gamma &=\\tan \\alpha \\tan \\beta \\tan \\gamma \\\\1&=\\cot \\beta \\cot \\gamma +\\cot \\gamma \\cot \\alpha +\\cot \\alpha \\cot \\beta \\\\\\cot \\left({\\frac {\\alpha }{2}}\\right)+\\cot \\left({\\frac {\\beta }{2}}\\right)+\\cot \\left({\\frac {\\gamma }{2}}\\right)&=\\cot \\left({\\frac {\\alpha }{2}}\\right)\\cot \\left({\\frac {\\beta }{2}}\\right)\\cot \\left({\\frac {\\gamma }{2}}\\right)\\\\1&=\\tan \\left({\\frac {\\beta }{2}}\\right)\\tan \\left({\\frac {\\gamma }{2}}\\right)+\\tan \\left({\\frac {\\gamma }{2}}\\right)\\tan \\left({\\frac {\\alpha }{2}}\\right)+\\tan \\left({\\frac {\\alpha }{2}}\\right)\\tan \\left({\\frac {\\beta }{2}}\\right)\\\\\\sin \\alpha +\\sin \\beta +\\sin \\gamma &=4\\cos \\left({\\frac {\\alpha }{2}}\\right)\\cos \\left({\\frac {\\beta }{2}}\\right)\\cos \\left({\\frac {\\gamma }{2}}\\right)\\\\-\\sin \\alpha +\\sin \\beta +\\sin \\gamma &=4\\cos \\left({\\frac {\\alpha }{2}}\\right)\\sin \\left({\\frac {\\beta }{2}}\\right)\\sin \\left({\\frac {\\gamma }{2}}\\right)\\\\\\cos \\alpha +\\cos \\beta +\\cos \\gamma &=4\\sin \\left({\\frac {\\alpha }{2}}\\right)\\sin \\left({\\frac {\\beta }{2}}\\right)\\sin \\left({\\frac {\\gamma }{2}}\\right)+1\\\\-\\cos \\alpha +\\cos \\beta +\\cos \\gamma &=4\\sin \\left({\\frac {\\alpha }{2}}\\right)\\cos \\left({\\frac {\\beta }{2}}\\right)\\cos \\left({\\frac {\\gamma }{2}}\\right)-1\\\\\\sin(2\\alpha )+\\sin(2\\beta )+\\sin(2\\gamma )&=4\\sin \\alpha \\sin \\beta \\sin \\gamma \\\\-\\sin(2\\alpha )+\\sin(2\\beta )+\\sin(2\\gamma )&=4\\sin \\alpha \\cos \\beta \\cos \\gamma \\\\\\cos(2\\alpha )+\\cos(2\\beta )+\\cos(2\\gamma )&=-4\\cos \\alpha \\cos \\beta \\cos \\gamma -1\\\\-\\cos(2\\alpha )+\\cos(2\\beta )+\\cos(2\\gamma )&=-4\\cos \\alpha \\sin \\beta \\sin \\gamma +1\\\\\\sin ^{2}\\alpha +\\sin ^{2}\\beta +\\sin ^{2}\\gamma &=2\\cos \\alpha \\cos \\beta \\cos \\gamma +2\\\\-\\sin ^{2}\\alpha +\\sin ^{2}\\beta +\\sin ^{2}\\gamma &=2\\cos \\alpha \\sin \\beta \\sin \\gamma \\\\\\cos ^{2}\\alpha +\\cos ^{2}\\beta +\\cos ^{2}\\gamma &=-2\\cos \\alpha \\cos \\beta \\cos \\gamma +1\\\\-\\cos ^{2}\\alpha +\\cos ^{2}\\beta +\\cos ^{2}\\gamma &=-2\\cos \\alpha \\sin \\beta \\sin \\gamma +1\\\\\\sin ^{2}(2\\alpha )+\\sin ^{2}(2\\beta )+\\sin ^{2}(2\\gamma )&=-2\\cos(2\\alpha )\\cos(2\\beta )\\cos(2\\gamma )+2\\\\\\cos ^{2}(2\\alpha )+\\cos ^{2}(2\\beta )+\\cos ^{2}(2\\gamma )&=2\\cos(2\\alpha )\\,\\cos(2\\beta )\\,\\cos(2\\gamma )+1\\\\1&=\\sin ^{2}\\left({\\frac {\\alpha }{2}}\\right)+\\sin ^{2}\\left({\\frac {\\beta }{2}}\\right)+\\sin ^{2}\\left({\\frac {\\gamma }{2}}\\right)+2\\sin \\left({\\frac {\\alpha }{2}}\\right)\\,\\sin \\left({\\frac {\\beta }{2}}\\right)\\,\\sin \\left({\\frac {\\gamma }{2}}\\right)\\end{aligned}}} 
## Historical shorthands
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=42 "Edit section: Historical shorthands")\]
Main articles: [Versine](https://en.wikipedia.org/wiki/Versine "Versine") and [Exsecant](https://en.wikipedia.org/wiki/Exsecant "Exsecant")
The [versine](https://en.wikipedia.org/wiki/Versine "Versine"), [coversine](https://en.wikipedia.org/wiki/Coversine "Coversine"), [haversine](https://en.wikipedia.org/wiki/Haversine "Haversine"), and [exsecant](https://en.wikipedia.org/wiki/Exsecant "Exsecant") were used in navigation. For example, the [haversine formula](https://en.wikipedia.org/wiki/Haversine_formula "Haversine formula") was used to calculate the distance between two points on a sphere. They are rarely used today.
## Miscellaneous
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=43 "Edit section: Miscellaneous")\]
### Dirichlet kernel
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=44 "Edit section: Dirichlet kernel")\]
Main article: [Dirichlet kernel](https://en.wikipedia.org/wiki/Dirichlet_kernel "Dirichlet kernel")
The **[Dirichlet kernel](https://en.wikipedia.org/wiki/Dirichlet_kernel "Dirichlet kernel")** *Dn*(*x*) is the function occurring on both sides of the next identity: 1 \+ 2 cos x \+ 2 cos ( 2 x ) \+ 2 cos ( 3 x ) \+ ⋯ \+ 2 cos ( n x ) \= sin ( ( n \+ 1 2 ) x ) sin ( 1 2 x ) . {\\displaystyle 1+2\\cos x+2\\cos(2x)+2\\cos(3x)+\\cdots +2\\cos(nx)={\\frac {\\sin \\left(\\left(n+{\\frac {1}{2}}\\right)x\\right)}{\\sin \\left({\\frac {1}{2}}x\\right)}}.} 
The [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") of any [integrable function](https://en.wikipedia.org/wiki/Integrable_function "Integrable function") of period 2 π {\\displaystyle 2\\pi }  with the Dirichlet kernel coincides with the function's n {\\displaystyle n} th-degree Fourier approximation. The same holds for any [measure](https://en.wikipedia.org/wiki/Measure_\(mathematics\) "Measure (mathematics)") or [generalized function](https://en.wikipedia.org/wiki/Distribution_\(mathematics\) "Distribution (mathematics)").
### Tangent half-angle substitution
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=45 "Edit section: Tangent half-angle substitution")\]
Main article: [Tangent half-angle substitution](https://en.wikipedia.org/wiki/Tangent_half-angle_substitution "Tangent half-angle substitution")
If we set t \= tan x 2 , {\\displaystyle t=\\tan {\\frac {x}{2}},}  then[\[60\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-60) sin x \= 2 t 1 \+ t 2 ; cos x \= 1 − t 2 1 \+ t 2 ; e i x \= 1 \+ i t 1 − i t ; d x \= 2 d t 1 \+ t 2 , {\\displaystyle \\sin x={\\frac {2t}{1+t^{2}}};\\qquad \\cos x={\\frac {1-t^{2}}{1+t^{2}}};\\qquad e^{ix}={\\frac {1+it}{1-it}};\\qquad dx={\\frac {2\\,dt}{1+t^{2}}},}  where e i x \= cos x \+ i sin x , {\\displaystyle e^{ix}=\\cos x+i\\sin x,}  sometimes abbreviated to [cis](https://en.wikipedia.org/wiki/Cis_\(mathematics\) "Cis (mathematics)") *x*.
When this substitution of t {\\displaystyle t}  for tan *x*/2 is used in [calculus](https://en.wikipedia.org/wiki/Calculus "Calculus"), it follows that sin x {\\displaystyle \\sin x}  is replaced by 2*t*/1 + *t*2, cos x {\\displaystyle \\cos x}  is replaced by 1 − *t*2/1 + *t*2 and the differential d*x* is replaced by 2 d*t*/1 + *t*2. Thereby one converts rational functions of sin x {\\displaystyle \\sin x}  and cos x {\\displaystyle \\cos x}  to rational functions of t {\\displaystyle t}  in order to find their [antiderivatives](https://en.wikipedia.org/wiki/Antiderivative "Antiderivative").
### Viète's infinite product
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=46 "Edit section: Viète's infinite product")\]
See also: [Viète's formula](https://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula "Viète's formula") and [Sinc function](https://en.wikipedia.org/wiki/Sinc_function "Sinc function")
cos θ 2 ⋅ cos θ 4 ⋅ cos θ 8 ⋯ \= ∏ n \= 1 ∞ cos θ 2 n \= sin θ θ \= sinc θ . {\\displaystyle \\cos {\\frac {\\theta }{2}}\\cdot \\cos {\\frac {\\theta }{4}}\\cdot \\cos {\\frac {\\theta }{8}}\\cdots =\\prod \_{n=1}^{\\infty }\\cos {\\frac {\\theta }{2^{n}}}={\\frac {\\sin \\theta }{\\theta }}=\\operatorname {sinc} \\theta .} 
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=47 "Edit section: See also")\]
- [Aristarchus's inequality](https://en.wikipedia.org/wiki/Aristarchus%27s_inequality "Aristarchus's inequality")
- [Derivatives of trigonometric functions](https://en.wikipedia.org/wiki/Table_of_derivatives#Derivatives_of_trigonometric_functions "Table of derivatives")
- [Exact trigonometric values](https://en.wikipedia.org/wiki/Exact_trigonometric_values "Exact trigonometric values") (values of sine and cosine expressed in surds)
- [Exsecant](https://en.wikipedia.org/wiki/Exsecant "Exsecant")
- [Half-side formula](https://en.wikipedia.org/wiki/Half-side_formula "Half-side formula")
- [Hyperbolic function](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function")
- Laws for solution of triangles:
- [Law of cosines](https://en.wikipedia.org/wiki/Law_of_cosines "Law of cosines")
- [Spherical law of cosines](https://en.wikipedia.org/wiki/Spherical_law_of_cosines "Spherical law of cosines")
- [Law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines")
- [Law of tangents](https://en.wikipedia.org/wiki/Law_of_tangents "Law of tangents")
- [Law of cotangents](https://en.wikipedia.org/wiki/Law_of_cotangents "Law of cotangents")
- [Mollweide's formula](https://en.wikipedia.org/wiki/Mollweide%27s_formula "Mollweide's formula")
- [List of integrals of trigonometric functions](https://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions "List of integrals of trigonometric functions")
- [Mnemonics in trigonometry](https://en.wikipedia.org/wiki/Mnemonics_in_trigonometry "Mnemonics in trigonometry")
- [Pentagramma mirificum](https://en.wikipedia.org/wiki/Pentagramma_mirificum "Pentagramma mirificum")
- [Proofs of trigonometric identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities "Proofs of trigonometric identities")
- [Prosthaphaeresis](https://en.wikipedia.org/wiki/Prosthaphaeresis "Prosthaphaeresis")
- [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem")
- [Tangent half-angle formula](https://en.wikipedia.org/wiki/Tangent_half-angle_formula "Tangent half-angle formula")
- [Trigonometric number](https://en.wikipedia.org/wiki/Trigonometric_number "Trigonometric number")
- [Trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry")
- [Uses of trigonometry](https://en.wikipedia.org/wiki/Uses_of_trigonometry "Uses of trigonometry")
- [Versine](https://en.wikipedia.org/wiki/Versine "Versine") and [haversine](https://en.wikipedia.org/wiki/Haversine "Haversine")
## References
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=48 "Edit section: References")\]
1. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-AS4345_1-0)**
[Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene Ann](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun"), eds. (1983) \[June 1964\]. ["Chapter 4, eqn 4.3.45"](http://www.math.ubc.ca/~cbm/aands/page_73.htm). [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun"). Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 73. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-61272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-61272-0 "Special:BookSources/978-0-486-61272-0")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [64-60036](https://lccn.loc.gov/64-60036). [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0167642](https://mathscinet.ams.org/mathscinet-getitem?mr=0167642). [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [65-12253](https://www.loc.gov/item/65012253).
2. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-2)** [Selby 1970](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#CITEREFSelby1970), p. 188
3. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-3)** Abramowitz and Stegun, p. 72, 4.3.13–15
4. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-4)** Abramowitz and Stegun, p. 72, 4.3.7–9
5. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-5)** Abramowitz and Stegun, p. 72, 4.3.16
6. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_addition_6-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_addition_6-1) [***c***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_addition_6-2) [***d***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_addition_6-3)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Trigonometric Addition Formulas"](https://mathworld.wolfram.com/TrigonometricAdditionFormulas.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
7. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-7)** Abramowitz and Stegun, p. 72, 4.3.17
8. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-8)** Abramowitz and Stegun, p. 72, 4.3.18
9. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-:0_9-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-:0_9-1)
["Angle Sum and Difference Identities"](http://www.milefoot.com/math/trig/22anglesumidentities.htm). *www.milefoot.com*. Retrieved 2019-10-12.
10. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-10)** Abramowitz and Stegun, p. 72, 4.3.19
11. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-11)** Abramowitz and Stegun, p. 80, 4.4.32
12. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-12)** Abramowitz and Stegun, p. 80, 4.4.33
13. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-13)** Abramowitz and Stegun, p. 80, 4.4.34
14. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-14)**
Bronstein, Manuel (1989). "Simplification of real elementary functions". In Gonnet, G. H. (ed.). *Proceedings of the ACM-[SIGSAM](https://en.wikipedia.org/wiki/SIGSAM "SIGSAM") 1989 International Symposium on Symbolic and Algebraic Computation*. ISSAC '89 (Portland US-OR, 1989-07). New York: [ACM](https://en.wikipedia.org/wiki/Association_for_Computing_Machinery "Association for Computing Machinery"). pp. 207–211\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1145/74540.74566](https://doi.org/10.1145%2F74540.74566). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-89791-325-6](https://en.wikipedia.org/wiki/Special:BookSources/0-89791-325-6 "Special:BookSources/0-89791-325-6")
.
15. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-15)** Michael Hardy. (2016). "On Tangents and Secants of Infinite Sums." *The American Mathematical Monthly*, volume 123, number 7, 701–703. <https://doi.org/10.4169/amer.math.monthly.123.7.701>
16. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-16)**
Hardy, Michael (2025). "Invariance of the Cauchy Family Under Linear Fractional Transformations". *The American Mathematical Monthly*. **132** (5): 453–455\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/00029890.2025.2459048](https://doi.org/10.1080%2F00029890.2025.2459048).
17. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-17)**
Knight, F. B. (1976). "A characterization of the Cauchy type". *Proceedings of the American Mathematical Society*. **1976**: 130–135\.
18. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-18)**
Hardy, Michael (2016). ["On Tangents and Secants of Infinite Sums"](https://zenodo.org/record/1000408). *American Mathematical Monthly*. **123** (7): 701–703\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.4169/amer.math.monthly.123.7.701](https://doi.org/10.4169%2Famer.math.monthly.123.7.701).
19. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-cut-the-knot.org_19-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-cut-the-knot.org_19-1)
["Sine, Cosine, and Ptolemy's Theorem"](https://www.cut-the-knot.org/proofs/sine_cosine.shtml).
20. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_multiple_angle_20-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_multiple_angle_20-1)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Multiple-Angle Formulas"](https://mathworld.wolfram.com/Multiple-AngleFormulas.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
21. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-21)** Abramowitz and Stegun, p. 74, 4.3.48
22. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-STM1_22-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-STM1_22-1) [Selby 1970](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#CITEREFSelby1970), pg. 190
23. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-A&S_23-0)** Cite error: The named reference `A&S` was invoked but never defined (see the [help page](https://en.wikipedia.org/wiki/Help:Cite_errors/Cite_error_references_no_text "Help:Cite errors/Cite error references no text")).
24. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-24)**
Weisstein, Eric W. ["Multiple-Angle Formulas"](https://mathworld.wolfram.com/Multiple-AngleFormulas.html). *mathworld.wolfram.com*. Retrieved 2022-02-06.
25. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-25)**
Ward, Ken. ["Multiple angles recursive formula"](http://trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm). *Ken Ward's Mathematics Pages*.
26. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-ReferenceA_26-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-ReferenceA_26-1)
[Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene Ann](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun"), eds. (1983) \[June 1964\]. ["Chapter 4, eqn 4.3.20-22"](http://www.math.ubc.ca/~cbm/aands/page_72.htm). [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun"). Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 72. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-61272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-61272-0 "Special:BookSources/978-0-486-61272-0")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [64-60036](https://lccn.loc.gov/64-60036). [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0167642](https://mathscinet.ams.org/mathscinet-getitem?mr=0167642). [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [65-12253](https://www.loc.gov/item/65012253).
27. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_half_angle_27-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_half_angle_27-1)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Half-Angle Formulas"](https://mathworld.wolfram.com/Half-AngleFormulas.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
28. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-28)** Abramowitz and Stegun, p. 72, 4.3.24–26
29. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_double_angle_29-0)**
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Double-Angle Formulas"](https://mathworld.wolfram.com/Double-AngleFormulas.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
30. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Stegun_p._72,_4_30-0)** Abramowitz and Stegun, p. 72, 4.3.27–28
31. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-31)** Abramowitz and Stegun, p. 72, 4.3.31–33
32. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-32)**
Eves, Howard (1990). *An introduction to the history of mathematics* (6th ed.). Philadelphia: Saunders College Pub. p. 309. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-03-029558-0](https://en.wikipedia.org/wiki/Special:BookSources/0-03-029558-0 "Special:BookSources/0-03-029558-0")
. [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [20842510](https://search.worldcat.org/oclc/20842510).
33. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-A&S_sum-to-product_33-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-A&S_sum-to-product_33-1) Abramowitz and Stegun, p. 72, 4.3.34–39
34. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-34)**
Johnson, Warren P. (Apr 2010). "Trigonometric Identities à la Hermite". *[American Mathematical Monthly](https://en.wikipedia.org/wiki/American_Mathematical_Monthly "American Mathematical Monthly")*. **117** (4): 311–327\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.4169/000298910x480784](https://doi.org/10.4169%2F000298910x480784). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [29690311](https://api.semanticscholar.org/CorpusID:29690311).
35. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-35)**
["Product Identity Multiple Angle"](https://math.stackexchange.com/q/2095330).
36. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-36)** Apostol, T.M. (1967) Calculus. 2nd edition. New York, NY, Wiley. Pp 334-335.
37. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-ReferenceB_37-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-ReferenceB_37-1)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Harmonic Addition Theorem"](https://mathworld.wolfram.com/HarmonicAdditionTheorem.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
38. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Muniz_38-0)**
Ortiz Muñiz, Eddie (Feb 1953). "A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities". *American Journal of Physics*. **21** (2): 140. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1953AmJPh..21..140M](https://ui.adsabs.harvard.edu/abs/1953AmJPh..21..140M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.1933371](https://doi.org/10.1119%2F1.1933371).
39. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-39)**
Agarwal, Ravi P.; O'Regan, Donal (2008). [*Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems*](https://books.google.com/books?id=jWvAfcNnphIC) (illustrated ed.). Springer Science & Business Media. p. 185. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-79146-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-79146-3 "Special:BookSources/978-0-387-79146-3")
.
[Extract of page 185](https://books.google.com/books?id=jWvAfcNnphIC&pg=PA185)
40. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-40)**
Jeffrey, Alan; Dai, Hui-hui (2008). "Section 2.4.1.6". *Handbook of Mathematical Formulas and Integrals* (4th ed.). Academic Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-12-374288-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-374288-9 "Special:BookSources/978-0-12-374288-9")
.
41. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-41)**
Fay, Temple H.; Kloppers, P. Hendrik (2001). ["The Gibbs' phenomenon"](https://dx.doi.org/10.1080/00207390117151). *International Journal of Mathematical Education in Science and Technology*. **32** (1): 73–89\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/00207390117151](https://doi.org/10.1080%2F00207390117151).
42. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-42)** Abramowitz and Stegun, p. 74, 4.3.47
43. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-43)** Abramowitz and Stegun, p. 71, 4.3.2
44. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-44)** Abramowitz and Stegun, p. 71, 4.3.1
45. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-45)** Abramowitz and Stegun, p. 80, 4.4.26–31
46. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-46)**
Hawkins, Faith Mary; Hawkins, J. Q. (March 1, 1969). [*Complex Numbers and Elementary Complex Functions*](https://archive.org/details/isbn_356025055/mode/2up). London: MacDonald Technical & Scientific London (published 1968). p. 122. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0356025056](https://en.wikipedia.org/wiki/Special:BookSources/978-0356025056 "Special:BookSources/978-0356025056")
.
47. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-47)**
Markushevich, A. I. (1966). [*The Remarkable Sine Function*](https://archive.org/details/markushevich-the-remarkable-sine-functions). New York: American Elsevier Publishing Company, Inc. pp. 35–37, 81. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1483256313](https://en.wikipedia.org/wiki/Special:BookSources/978-1483256313 "Special:BookSources/978-1483256313")
.
48. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-48)** Abramowitz and Stegun, p. 74, 4.3.65–66
49. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-49)** Abramowitz and Stegun, p. 75, 4.3.89–90
50. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-50)** Abramowitz and Stegun, p. 85, 4.5.68–69
51. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-51)** [Abramowitz & Stegun 1972](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#CITEREFAbramowitzStegun1972), p. 73, 4.3.45
52. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Wu_52-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Wu_52-1) [***c***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Wu_52-2) Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", *Mathematics Magazine* 77(3), June 2004, p. 189.
53. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-53)**
S. M. Abrarov; R. K. Jagpal; R. Siddiqui; B. M. Quine (2021), "Algorithmic determination of a large integer in the two-term Machin-like formula for π", *Mathematics*, **9** (17), 2162, [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[2107\.01027](https://arxiv.org/abs/2107.01027), [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3390/math9172162](https://doi.org/10.3390%2Fmath9172162)
54. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-54)**
Humble, Steve (Nov 2004). "Grandma's identity". *Mathematical Gazette*. **88**: 524–525\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/s0025557200176223](https://doi.org/10.1017%2Fs0025557200176223). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [125105552](https://api.semanticscholar.org/CorpusID:125105552).
55. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-55)**
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Sine"](https://mathworld.wolfram.com/Sine.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
56. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Harris_56-0)** Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, *Proofs Without Words* (1993, Mathematical Association of America), p. 39.
57. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-57)** [Milton Abramowitz and Irene Stegun, *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun"), [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"), New York, 1972, formulae 9.1.42–9.1.45
58. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-58)** Er. K. C. Joshi, *Krishna's IIT MATHEMATIKA*. Krishna Prakashan Media. Meerut, India. page 636.
59. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-59)** Cagnoli, Antonio (1808), *Trigonométrie rectiligne et sphérique*, p. 27.
60. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-60)** Abramowitz and Stegun, p. 72, 4.3.23
## Bibliography
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=49 "Edit section: Bibliography")\]
- [Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene A.](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun"), eds. (1972). [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://archive.org/details/handbookofmathe000abra). New York: [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-61272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-61272-0 "Special:BookSources/978-0-486-61272-0")
.
- Nielsen, Kaj L. (1966), *Logarithmic and Trigonometric Tables to Five Places* (2nd ed.), New York: [Barnes & Noble](https://en.wikipedia.org/wiki/Barnes_%26_Noble "Barnes & Noble"), [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [61-9103](https://lccn.loc.gov/61-9103)
- Selby, Samuel M., ed. (1970), *Standard Mathematical Tables* (18th ed.), The Chemical Rubber Co.
## External links
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=50 "Edit section: External links")\]
- [Values of sin and cos, expressed in surds, for integer multiples of 3° and of 5+5/8°](http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html)
, and for the same angles [csc and sec](http://www.jdawiseman.com/papers/easymath/surds_csc_sec.html) and [tan](http://www.jdawiseman.com/papers/easymath/surds_tan.html)
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List of trigonometric identities
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| Readable Markdown | In [trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry"), **trigonometric identities** are [equalities](https://en.wikipedia.org/wiki/Equality_\(mathematics\) "Equality (mathematics)") that involve [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions") and are true for every value of the occurring [variables](https://en.wikipedia.org/wiki/Variable_\(mathematics\) "Variable (mathematics)") for which both sides of the equality are defined. Geometrically, these are [identities](https://en.wikipedia.org/wiki/Identity_\(mathematics\) "Identity (mathematics)") involving certain functions of one or more [angles](https://en.wikipedia.org/wiki/Angle "Angle"). They are distinct from [triangle identities](https://en.wikipedia.org/wiki/Trigonometry#Triangle_identities "Trigonometry"), which are identities potentially involving angles but also involving side lengths or other lengths of a [triangle](https://en.wikipedia.org/wiki/Triangle "Triangle").
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the [integration](https://en.wikipedia.org/wiki/Integral "Integral") of non-trigonometric functions: a common technique involves first using the [substitution rule with a trigonometric function](https://en.wikipedia.org/wiki/Trigonometric_substitution "Trigonometric substitution"), and then simplifying the resulting integral with a trigonometric identity.
## Pythagorean identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=1 "Edit section: Pythagorean identities")\]
[](https://en.wikipedia.org/wiki/File:Trigonometric_functions_and_their_reciprocals_on_the_unit_circle.svg)
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity , and the red triangle shows that .
The basic relationship between the [sine and cosine](https://en.wikipedia.org/wiki/Sine_and_cosine "Sine and cosine") is given by the Pythagorean identity:
where  means  and  means 
This can be viewed as a version of the [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem"), and follows from the equation  for the [unit circle](https://en.wikipedia.org/wiki/Unit_circle "Unit circle"). This equation can be solved for either the sine or the cosine:
where the sign depends on the [quadrant](https://en.wikipedia.org/wiki/Quadrant_\(plane_geometry\) "Quadrant (plane geometry)") of 
Dividing this identity by , , or both yields the following identities: 
Using these identities, it is possible to express any trigonometric function in terms of any other ([up to](https://en.wikipedia.org/wiki/Up_to "Up to") a plus or minus sign):
| in terms of |  |
|---|---|
## Reflections, shifts, and periodicity
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=2 "Edit section: Reflections, shifts, and periodicity")\]
By examining the unit circle, one can establish the following properties of the trigonometric functions.
[](https://en.wikipedia.org/wiki/File:Unit_Circle_-_symmetry.svg)
Transformation of coordinates (*a*,*b*) when shifting the reflection angle  in increments of 
When the direction of a [Euclidean vector](https://en.wikipedia.org/wiki/Euclidean_vector "Euclidean vector") is represented by an angle  this is the angle determined by the free vector (starting at the origin) and the positive \-unit vector. The same concept may also be applied to lines in an [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space "Euclidean space"), where the angle is that determined by a parallel to the given line through the origin and the positive \-axis. If a line (vector) with direction  is reflected about a line with direction  then the direction angle  of this reflected line (vector) has the value 
The values of the trigonometric functions of these angles  for specific angles  satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as *reduction formulae*.[\[2\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-2)
|  reflected in [\[3\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-3) [odd/even](https://en.wikipedia.org/wiki/Even_and_odd_functions "Even and odd functions") identities |
|---|
### Shifts and periodicity
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=4 "Edit section: Shifts and periodicity")\]
[](https://en.wikipedia.org/wiki/File:Unit_Circle_-_shifts.svg)
Transformation of coordinates (*a*,*b*) when shifting the angle  in increments of 
| Shift by one quarter period | Shift by one half period | Shift by full periods[\[4\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-4) | Period |
|---|---|---|---|
|  | | | |
The sign of trigonometric functions depends on quadrant of the angle. If  and sgn is the [sign function](https://en.wikipedia.org/wiki/Sign_function "Sign function"),
![{\\displaystyle {\\begin{aligned}\\operatorname {sgn} (\\sin \\theta )=\\operatorname {sgn} (\\csc \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ 0\<\\theta \<\\pi \\\\-1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<0\\\\0&{\\text{if}}\\ \\ \\theta \\in \\{0,\\pi \\}\\end{cases}}\\\\\[5mu\]\\operatorname {sgn} (\\cos \\theta )=\\operatorname {sgn} (\\sec \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ {-{\\tfrac {\\pi }{2}}}\<\\theta \<{\\tfrac {\\pi }{2}}\\\\-1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<-{\\tfrac {\\pi }{2}}\\ \\ {\\text{or}}\\ \\ {\\tfrac {\\pi }{2}}\<\\theta \<\\pi \\\\0&{\\text{if}}\\ \\ \\theta \\in {\\bigl \\{}{-{\\tfrac {\\pi }{2}}},{\\tfrac {\\pi }{2}}{\\bigr \\}}\\end{cases}}\\\\\[5mu\]\\operatorname {sgn} (\\tan \\theta )=\\operatorname {sgn} (\\cot \\theta )&={\\begin{cases}+1&{\\text{if}}\\ \\ {-\\pi }\<\\theta \<-{\\tfrac {\\pi }{2}}\\ \\ {\\text{or}}\\ \\ 0\<\\theta \<{\\tfrac {\\pi }{2}}\\\\-1&{\\text{if}}\\ \\ {-{\\tfrac {\\pi }{2}}}\<\\theta \<0\\ \\ {\\text{or}}\\ \\ {\\tfrac {\\pi }{2}}\<\\theta \<\\pi \\\\0&{\\text{if}}\\ \\ \\theta \\in {\\bigl \\{}{-{\\tfrac {\\pi }{2}}},0,{\\tfrac {\\pi }{2}},\\pi {\\bigr \\}}\\end{cases}}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3885aee3a70a73da0b369ad2f21a8122f4e78587)
The trigonometric functions are periodic with common period  so for values of θ outside the interval ![{\\displaystyle ({-\\pi },\\pi \],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/122328145a5f3a4162d21b4fbb5f4d1149932d2b) they take repeating values (see [§ Shifts and periodicity](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Shifts_and_periodicity) above). The sign of a sinusoid or cosinusoid can be used to define a normalized [square wave](https://en.wikipedia.org/wiki/Square_wave_\(waveform\) "Square wave (waveform)"). For example, the functions  and  take values ±1 and correspond to square waves with a phase shift of *π*/2.
## Angle sum and difference identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=6 "Edit section: Angle sum and difference identities")\]
[](https://en.wikipedia.org/wiki/File:Angle_sum.svg)
Geometric construction to derive angle sum trigonometric identities
[](https://en.wikipedia.org/wiki/File:Diagram_showing_the_angle_difference_trigonometry_identities_for_sin\(a-b\)_and_cos\(a-b\).svg)
Diagram showing the angle difference identities for  and 
These are also known as the *angle addition and subtraction theorems* (or *formulae*). 
The angle difference identities for  and  can be derived from the angle sum versions (and vice versa) by substituting  for  and using the facts that  and  They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here. They can also be seen as expressing the [dot product](https://en.wikipedia.org/wiki/Dot_product "Dot product") and [cross product](https://en.wikipedia.org/wiki/Cross_product "Cross product") of two vectors in terms of the cosine and the sine of the angle between them.
These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.
| | |
|---|---|
| Sine |  |
### Sines and cosines of sums of infinitely many angles
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=7 "Edit section: Sines and cosines of sums of infinitely many angles")\]
When the series  [converges absolutely](https://en.wikipedia.org/wiki/Absolute_convergence "Absolute convergence") then
Because the series  converges absolutely, it is necessarily the case that   and  Particularly, in these two identities, an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are [cofinitely](https://en.wikipedia.org/wiki/Cofiniteness "Cofiniteness") many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.
When only finitely many of the angles  are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
### Tangents and cotangents of sums
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=8 "Edit section: Tangents and cotangents of sums")\]
Let  (for ) be the kth-degree [elementary symmetric polynomial](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial "Elementary symmetric polynomial") in the variables  for  that is,
![{\\displaystyle {\\begin{aligned}e\_{0}&=1\\\\\[6pt\]e\_{1}&=\\sum \_{i}x\_{i}&&=\\sum \_{i}\\tan \\theta \_{i}\\\\\[6pt\]e\_{2}&=\\sum \_{i\<j}x\_{i}x\_{j}&&=\\sum \_{i\<j}\\tan \\theta \_{i}\\tan \\theta \_{j}\\\\\[6pt\]e\_{3}&=\\sum \_{i\<j\<k}x\_{i}x\_{j}x\_{k}&&=\\sum \_{i\<j\<k}\\tan \\theta \_{i}\\tan \\theta \_{j}\\tan \\theta \_{k}\\\\&\\ \\ \\vdots &&\\ \\ \\vdots \\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1927ae2be96f5156e18813a20486fdb0ff72f1)
Then
 This can be shown by using the sine and cosine sum formulae above: ![{\\displaystyle {\\begin{aligned}\\tan {\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {{\\sin }{\\bigl (}\\sum \_{i}\\theta \_{i}{\\bigr )}/\\prod \_{i}\\cos \\theta \_{i}}{{\\cos }{\\bigl (}\\sum \_{i}\\theta \_{i}{\\bigr )}/\\prod \_{i}\\cos \\theta \_{i}}}\\\\\[10pt\]&={\\frac {\\displaystyle \\sum \_{{\\text{odd}}\\ k\\geq 1}(-1)^{\\frac {k-1}{2}}\\sum \_{\\begin{smallmatrix}A\\subseteq \\{1,2,3,\\dots \\}\\\\\\left\|A\\right\|=k\\end{smallmatrix}}\\prod \_{i\\in A}\\tan \\theta \_{i}}{\\displaystyle \\sum \_{{\\text{even}}\\ k\\geq 0}~(-1)^{\\frac {k}{2}}\~~\\sum \_{\\begin{smallmatrix}A\\subseteq \\{1,2,3,\\dots \\}\\\\\\left\|A\\right\|=k\\end{smallmatrix}}\\prod \_{i\\in A}\\tan \\theta \_{i}}}={\\frac {e\_{1}-e\_{3}+e\_{5}-\\cdots }{e\_{0}-e\_{2}+e\_{4}-\\cdots }}\\\\\[10pt\]\\cot {\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {e\_{0}-e\_{2}+e\_{4}-\\cdots }{e\_{1}-e\_{3}+e\_{5}-\\cdots }}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ff3a24e1e073c731c170651eca41efdad72212)
The number of terms on the right side depends on the number of terms on the left side.
For example: ![{\\displaystyle {\\begin{aligned}\\tan(\\theta \_{1}+\\theta \_{2})&={\\frac {e\_{1}}{e\_{0}-e\_{2}}}={\\frac {x\_{1}+x\_{2}}{1\\ -\\ x\_{1}x\_{2}}}={\\frac {\\tan \\theta \_{1}+\\tan \\theta \_{2}}{1\\ -\\ \\tan \\theta \_{1}\\tan \\theta \_{2}}},\\\\\[8pt\]\\tan(\\theta \_{1}+\\theta \_{2}+\\theta \_{3})&={\\frac {e\_{1}-e\_{3}}{e\_{0}-e\_{2}}}={\\frac {(x\_{1}+x\_{2}+x\_{3})\\ -\\ (x\_{1}x\_{2}x\_{3})}{1\\ -\\ (x\_{1}x\_{2}+x\_{1}x\_{3}+x\_{2}x\_{3})}},\\\\\[8pt\]\\tan(\\theta \_{1}+\\theta \_{2}+\\theta \_{3}+\\theta \_{4})&={\\frac {e\_{1}-e\_{3}}{e\_{0}-e\_{2}+e\_{4}}}\\\\\[8pt\]&={\\frac {(x\_{1}+x\_{2}+x\_{3}+x\_{4})\\ -\\ (x\_{1}x\_{2}x\_{3}+x\_{1}x\_{2}x\_{4}+x\_{1}x\_{3}x\_{4}+x\_{2}x\_{3}x\_{4})}{1\\ -\\ (x\_{1}x\_{2}+x\_{1}x\_{3}+x\_{1}x\_{4}+x\_{2}x\_{3}+x\_{2}x\_{4}+x\_{3}x\_{4})\\ +\\ (x\_{1}x\_{2}x\_{3}x\_{4})}},\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9196b8ab5e0aef6cf7d7fec7d7546f2a6e7d2f)
and so on. The case of only finitely many terms can be proved by [mathematical induction](https://en.wikipedia.org/wiki/Mathematical_induction "Mathematical induction").[\[14\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-14) The case of infinitely many terms can be proved by using some elementary inequalities.[\[15\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-15)
Suppose  and  and
and let  be any number for which  Suppose that  so that the forgoing fraction cannot be 0/0. Then for all [\[16\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-16)
(In case the denominator of this fraction is 0, we take the value of the fraction to be , where the symbol  does not mean either  or , but is the  that is approached by going in either the positive or the negative direction, making the completion of the line  topologically a circle.)
From this identity it can be shown to follow quickly that the family of all [Cauchy-distributed](https://en.wikipedia.org/wiki/Cauchy_distribution "Cauchy distribution") random variables is closed under linear fractional transformations, a result known since 1976.[\[17\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-17)
### Secants and cosecants of sums
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=10 "Edit section: Secants and cosecants of sums")\]
![{\\displaystyle {\\begin{aligned}{\\sec }{\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {\\prod \_{i}\\sec \\theta \_{i}}{e\_{0}-e\_{2}+e\_{4}-\\cdots }}\\\\\[8pt\]{\\csc }{\\Bigl (}\\sum \_{i}\\theta \_{i}{\\Bigr )}&={\\frac {\\prod \_{i}\\sec \\theta \_{i}}{e\_{1}-e\_{3}+e\_{5}-\\cdots }}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b24f2b169fbe11608507c63f5bb728e03b2b4957)
where  is the kth-degree [elementary symmetric polynomial](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial "Elementary symmetric polynomial") in the n variables   and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.[\[18\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-18) The case of only finitely many terms can be proved by mathematical induction on the number of such terms.
For example,
![{\\displaystyle {\\begin{aligned}\\sec(\\alpha +\\beta +\\gamma )&={\\frac {\\sec \\alpha \\sec \\beta \\sec \\gamma }{1-\\tan \\alpha \\tan \\beta -\\tan \\alpha \\tan \\gamma -\\tan \\beta \\tan \\gamma }}\\\\\[8pt\]\\csc(\\alpha +\\beta +\\gamma )&={\\frac {\\sec \\alpha \\sec \\beta \\sec \\gamma }{\\tan \\alpha +\\tan \\beta +\\tan \\gamma -\\tan \\alpha \\tan \\beta \\tan \\gamma }}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e082e69aab7a9c782f590f3a93262ba3101899b7)
[](https://en.wikipedia.org/wiki/File:Diagram_illustrating_the_relation_between_Ptolemy%27s_theorem_and_the_angle_sum_trig_identity_for_sin.svg)
Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine. Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(*α* + *β*) = sin *α* cos *β* + cos *α* sin *β*.
Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a [cyclic quadrilateral](https://en.wikipedia.org/wiki/Cyclic_quadrilateral "Cyclic quadrilateral") , as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.[\[19\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-cut-the-knot.org-19) The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.
By [Thales's theorem](https://en.wikipedia.org/wiki/Thales%27s_theorem "Thales's theorem"),  and  are both right angles. The right-angled triangles  and  both share the hypotenuse  of length 1. Thus, the side , ,  and .
By the [inscribed angle](https://en.wikipedia.org/wiki/Inscribed_angle "Inscribed angle") theorem, the [central angle](https://en.wikipedia.org/wiki/Central_angle "Central angle") subtended by the chord  at the circle's center is twice the angle , i.e. . Therefore, the symmetrical pair of red triangles each has the angle  at the center. Each of these triangles has a [hypotenuse](https://en.wikipedia.org/wiki/Hypotenuse "Hypotenuse") of length , so the length of  is , i.e. simply . The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also .
When these values are substituted into the statement of Ptolemy's theorem that , this yields the angle sum trigonometric identity for sine: . The angle difference formula for  can be similarly derived by letting the side  serve as a diameter instead of .[\[19\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-cut-the-knot.org-19)
## Multiple-angle and half-angle formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=12 "Edit section: Multiple-angle and half-angle formulas")\]
| | |
|---|---|
| Tn is the nth [Chebyshev polynomial](https://en.wikipedia.org/wiki/Chebyshev_polynomials "Chebyshev polynomials") | [\[20\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-20) |
### Multiple-angle formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=13 "Edit section: Multiple-angle formulae")\]
#### Double-angle formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=14 "Edit section: Double-angle formulae")\]
[](https://en.wikipedia.org/wiki/File:Visual_demonstration_of_the_double-angle_trigonometric_identity_for_sine.svg)
Visual demonstration of the double-angle formula for sine. For the above isosceles triangle with unit sides and angle , the area 1/2 × base × height is calculated in two orientations. When upright, the area is . When on its side, the same area is . Therefore, 
Formulae for twice an angle.[\[22\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-STM1-22)[\[23\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-A&S-23)
#### Triple-angle formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=15 "Edit section: Triple-angle formulae")\]
Formulae for triple angles.[\[22\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-STM1-22)
#### Multiple-angle formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=16 "Edit section: Multiple-angle formulae")\]
Formulae for multiple angles.[\[24\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-24)
The [Chebyshev](https://en.wikipedia.org/wiki/Pafnuty_Chebyshev "Pafnuty Chebyshev") method is a [recursive](https://en.wikipedia.org/wiki/Recursion "Recursion") [algorithm](https://en.wikipedia.org/wiki/Algorithm "Algorithm") for finding the nth multiple angle formula knowing the th and th values.[\[25\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-25)
 can be computed from , , and  with
This can be proved by adding together the formulae
It follows by induction that  is a polynomial of  the so-called Chebyshev polynomial of the first kind, see [Chebyshev polynomials\#Trigonometric definition](https://en.wikipedia.org/wiki/Chebyshev_polynomials#Trigonometric_definition "Chebyshev polynomials").
Similarly,  can be computed from   and  with 
This can be proved by adding formulae for  and 
Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:
### Half-angle formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=18 "Edit section: Half-angle formulas")\]
![{\\displaystyle {\\begin{aligned}\\sin {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\sin {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {1-\\cos \\theta }{2}}}\\\\\[3pt\]\\cos {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\cos {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {1+\\cos \\theta }{2}}}\\\\\[3pt\]\\tan {\\frac {\\theta }{2}}&={\\frac {1-\\cos \\theta }{\\sin \\theta }}={\\frac {\\sin \\theta }{1+\\cos \\theta }}=\\csc \\theta -\\cot \\theta ={\\frac {\\tan \\theta }{1+\\sec {\\theta }}}\\\\\[6mu\]&=\\operatorname {sgn} (\\sin \\theta ){\\sqrt {\\frac {1-\\cos \\theta }{1+\\cos \\theta }}}={\\frac {-1+\\operatorname {sgn} (\\cos \\theta ){\\sqrt {1+\\tan ^{2}\\theta }}}{\\tan \\theta }}\\\\\[3pt\]\\cot {\\frac {\\theta }{2}}&={\\frac {1+\\cos \\theta }{\\sin \\theta }}={\\frac {\\sin \\theta }{1-\\cos \\theta }}=\\csc \\theta +\\cot \\theta =\\operatorname {sgn} (\\sin \\theta ){\\sqrt {\\frac {1+\\cos \\theta }{1-\\cos \\theta }}}\\\\\\sec {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\cos {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {2}{1+\\cos \\theta }}}\\\\\\csc {\\frac {\\theta }{2}}&=\\operatorname {sgn} \\left(\\sin {\\frac {\\theta }{2}}\\right){\\sqrt {\\frac {2}{1-\\cos \\theta }}}\\\\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d53d104c73c4b28f1dd7bfa5d2b0c7207c68cd87) [\[26\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-ReferenceA-26)[\[27\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_half_angle-27)
Also ![{\\displaystyle {\\begin{aligned}\\tan {\\frac {\\eta \\pm \\theta }{2}}&={\\frac {\\sin \\eta \\pm \\sin \\theta }{\\cos \\eta +\\cos \\theta }}\\\\\[3pt\]\\tan \\left({\\frac {\\theta }{2}}+{\\frac {\\pi }{4}}\\right)&=\\sec \\theta +\\tan \\theta \\\\\[3pt\]{\\sqrt {\\frac {1-\\sin \\theta }{1+\\sin \\theta }}}&={\\frac {\\left\|1-\\tan {\\frac {\\theta }{2}}\\right\|}{\\left\|1+\\tan {\\frac {\\theta }{2}}\\right\|}}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/639270245e163f1afa8682ac8aa12086db43dce1)
These can be shown by using either the sum and difference identities or the multiple-angle formulae.
| | Sine | Cosine | Tangent | Cotangent |
|---|---|---|---|---|
| Double-angle formula[\[28\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-28)[\[29\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_double_angle-29) |  | | | |
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a [compass and straightedge construction](https://en.wikipedia.org/wiki/Compass_and_straightedge_constructions "Compass and straightedge constructions") of [angle trisection](https://en.wikipedia.org/wiki/Angle_trisection "Angle trisection") to the algebraic problem of solving a [cubic equation](https://en.wikipedia.org/wiki/Cubic_function "Cubic function"), which allows one to prove that [trisection is in general impossible](https://en.wikipedia.org/wiki/Angle_trisection#Proof_of_impossibility "Angle trisection") using the given tools.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the [cubic equation](https://en.wikipedia.org/wiki/Cubic_function "Cubic function") 4*x*3 − 3*x* + *d* = 0, where x is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the [discriminant](https://en.wikipedia.org/wiki/Discriminant "Discriminant") of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). [None of these solutions are reducible](https://en.wikipedia.org/wiki/Casus_irreducibilis "Casus irreducibilis") to a real [algebraic expression](https://en.wikipedia.org/wiki/Algebraic_expression "Algebraic expression"), as they use intermediate complex numbers under the [cube roots](https://en.wikipedia.org/wiki/Cube_root "Cube root").
## Power-reduction formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=20 "Edit section: Power-reduction formulae")\]
Obtained by solving the second and third versions of the cosine double-angle formula.
| Sine | Cosine | Other |
|---|---|---|
|  | | |
In general terms of powers of  or  the following is true, and can be deduced using [De Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula "De Moivre's formula"), [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") and the [binomial theorem](https://en.wikipedia.org/wiki/Binomial_theorem "Binomial theorem").
| if *n* is ... |  |
|---|---|
## Product-to-sum and sum-to-product identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=21 "Edit section: Product-to-sum and sum-to-product identities")\]
[](https://en.wikipedia.org/wiki/File:Visual_proof_prosthaphaeresis_cosine_formula.svg)
Proof of the sum-and-difference-to-product cosine identity for prosthaphaeresis calculations using an [isosceles triangle](https://en.wikipedia.org/wiki/Isosceles_triangle "Isosceles triangle")
The product-to-sum identities[\[31\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-31) or [prosthaphaeresis](https://en.wikipedia.org/wiki/Prosthaphaeresis "Prosthaphaeresis") formulae can be proven by expanding their right-hand sides using the [angle addition theorems](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities). Historically, the first four of these were known as **Werner's formulas**, after [Johannes Werner](https://en.wikipedia.org/wiki/Johannes_Werner "Johannes Werner") who used them for astronomical calculations.[\[32\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-32) See [amplitude modulation](https://en.wikipedia.org/wiki/Amplitude_modulation#Simplified_analysis_of_standard_AM "Amplitude modulation") for an application of the product-to-sum formulae, and [beat (acoustics)](https://en.wikipedia.org/wiki/Beat_\(acoustics\) "Beat (acoustics)") and [phase detector](https://en.wikipedia.org/wiki/Phase_detector "Phase detector") for applications of the sum-to-product formulae.
### Product-to-sum identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=22 "Edit section: Product-to-sum identities")\]
The product of two sines or cosines of different angles can be converted to a sum of trigonometric functions of a sum and difference of those angles:
![{\\displaystyle {\\begin{aligned}\\cos \\theta \\,\\cos \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cos(\\theta -\\varphi )+\\cos(\\theta +\\varphi ){\\bigr )},\\\\\[5mu\]\\sin \\theta \\,\\sin \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cos(\\theta -\\varphi )-\\cos(\\theta +\\varphi ){\\bigr )},\\\\\[5mu\]\\sin \\theta \\,\\cos \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sin(\\theta +\\varphi )+\\sin(\\theta -\\varphi ){\\bigr )},\\\\\[5mu\]\\cos \\theta \\,\\sin \\varphi &={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sin(\\theta +\\varphi )-\\sin(\\theta -\\varphi ){\\bigr )}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd715f4b6380798a701537d9c067f28cc371efa) As a corollary, the product or quotient of tangents can be converted to a quotient of sums of cosines or sines, respectively, ![{\\displaystyle {\\begin{aligned}\\tan \\theta \\,\\tan \\varphi &={\\frac {\\cos(\\theta -\\varphi )-\\cos(\\theta +\\varphi )}{\\cos(\\theta -\\varphi )+\\cos(\\theta +\\varphi )}},\\\\\[5mu\]{\\frac {\\tan \\theta }{\\tan \\varphi }}&={\\frac {\\sin(\\theta +\\varphi )+\\sin(\\theta -\\varphi )}{\\sin(\\theta +\\varphi )-\\sin(\\theta -\\varphi )}}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69a3a5dee2394b1ed85551729df20f0abd6770b7)
More generally, for a product of any number of sines or cosines,\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] ![{\\displaystyle {\\begin{aligned}\\prod \_{k=1}^{n}\\cos \\theta \_{k}&={\\frac {1}{2^{n}}}\\sum \_{e\\in S}\\cos(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\\\\[5mu\]&{\\text{where }}e=(e\_{1},\\ldots ,e\_{n})\\in S=\\{1,-1\\}^{n},\\\\\\prod \_{k=1}^{n}\\sin \\theta \_{k}&={\\frac {(-1)^{\\left\\lfloor {\\frac {n}{2}}\\right\\rfloor }}{2^{n}}}{\\begin{cases}\\displaystyle \\sum \_{e\\in S}\\cos(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\prod \_{j=1}^{n}e\_{j}\\;{\\text{if}}\\;n\\;{\\text{is even}},\\\\\\displaystyle \\sum \_{e\\in S}\\sin(e\_{1}\\theta \_{1}+\\cdots +e\_{n}\\theta \_{n})\\prod \_{j=1}^{n}e\_{j}\\;{\\text{if}}\\;n\\;{\\text{is odd}}.\\end{cases}}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/106ce6115e77f036a95a2096bc66b01927e5dff9)
### Sum-to-product identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=23 "Edit section: Sum-to-product identities")\]
[](https://en.wikipedia.org/wiki/File:Diagram_illustrating_sum_to_product_identities_for_sine_and_cosine.svg)
Diagram illustrating sum-to-product identities for sine and cosine. The blue right-angled triangle has angle  and the red right-angled triangle has angle . Both have a hypotenuse of length 1. Auxiliary angles, here called  and , are constructed such that  and . Therefore,  and . This allows the two congruent purple-outline triangles  and  to be constructed, each with hypotenuse  and angle  at their base. The sum of the heights of the red and blue triangles is , and this is equal to twice the height of one purple triangle, i.e. . Writing  and  in that equation in terms of  and  yields a sum-to-product identity for sine: . Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.
The sum of sines or cosines of two angles can be converted to a product of sines or cosines of the mean and half the difference of the angles:[\[33\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-A&S_sum-to-product-33)
![{\\displaystyle {\\begin{aligned}\\sin \\theta +\\sin \\varphi &=2\\sin {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\cos {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\sin \\theta -\\sin \\varphi &=2\\cos {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\sin {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\cos \\theta +\\cos \\varphi &=2\\cos {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\cos {\\tfrac {1}{2}}(\\theta -\\varphi ),\\\\\[5mu\]\\cos \\theta -\\cos \\varphi &=-2\\sin {\\tfrac {1}{2}}(\\theta +\\varphi )\\,\\sin {\\tfrac {1}{2}}(\\theta -\\varphi ).\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3676cbeab42be735d5fb198d843eeec23ee9b763)
The sum of the tangent of two angles can be converted to a quotient of the sine of angles divided by the product of the cosines:[\[33\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-A&S_sum-to-product-33) 
### Hermite's cotangent identity
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=24 "Edit section: Hermite's cotangent identity")\]
[Charles Hermite](https://en.wikipedia.org/wiki/Charles_Hermite "Charles Hermite") demonstrated the following identity.[\[34\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-34) Suppose  are [complex numbers](https://en.wikipedia.org/wiki/Complex_number "Complex number"), no two of which differ by an integer multiple of π. Let
(in particular,  being an [empty product](https://en.wikipedia.org/wiki/Empty_product "Empty product"), is 1). Then
The simplest non-trivial example is the case *n* = 2:
### Finite products of trigonometric functions
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=25 "Edit section: Finite products of trigonometric functions")\]
For [coprime](https://en.wikipedia.org/wiki/Coprime "Coprime") integers n, m
where Tn is the [Chebyshev polynomial](https://en.wikipedia.org/wiki/Chebyshev_polynomial "Chebyshev polynomial").\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
The following relationship holds for the sine function
More generally for an integer *n* \> 0[\[35\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-35)
or written in terms of the [chord](https://en.wikipedia.org/wiki/Chord_\(geometry\) "Chord (geometry)") function ,
This comes from the [factorization of the polynomial](https://en.wikipedia.org/wiki/Factorization_of_polynomials "Factorization of polynomials")  into linear factors (cf. [root of unity](https://en.wikipedia.org/wiki/Root_of_unity "Root of unity")): For any complex z and an integer *n* \> 0,
## Linear combinations
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=26 "Edit section: Linear combinations")\]
For some purposes it is important to know that any [linear combination](https://en.wikipedia.org/wiki/Linear_combination "Linear combination") of sine waves of the same period or frequency but different [phase shifts](https://en.wikipedia.org/wiki/Phase_\(waves\) "Phase (waves)") is also a sine wave with the same period or frequency, but a different phase shift. This is useful in [sinusoid](https://en.wikipedia.org/wiki/Sinusoid "Sinusoid") [data fitting](https://en.wikipedia.org/wiki/Data_fitting "Data fitting"), because the measured or observed data are linearly related to the a and b unknowns of the [in-phase and quadrature components](https://en.wikipedia.org/wiki/In-phase_and_quadrature_components "In-phase and quadrature components") basis below, resulting in a simpler [Jacobian](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant "Jacobian matrix and determinant"), compared to that of  and .
The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,[\[36\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-36)[\[37\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-ReferenceB-37)
where  and  are defined as so:
given that 
### Arbitrary phase shift
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=28 "Edit section: Arbitrary phase shift")\]
More generally, for arbitrary phase shifts, we have
where  and  satisfy:
### More than two sinusoids
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=29 "Edit section: More than two sinusoids")\]
The general case reads[\[37\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-ReferenceB-37)
 where  and 
## Lagrange's trigonometric identities
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=30 "Edit section: Lagrange's trigonometric identities")\]
These identities, named after [Joseph Louis Lagrange](https://en.wikipedia.org/wiki/Joseph_Louis_Lagrange "Joseph Louis Lagrange"), are:[\[38\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Muniz-38)[\[39\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-39)[\[40\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-40) ![{\\displaystyle {\\begin{aligned}\\sum \_{k=0}^{n}\\sin k\\theta &={\\frac {\\cos {\\tfrac {1}{2}}\\theta -\\cos \\left(\\left(n+{\\tfrac {1}{2}}\\right)\\theta \\right)}{2\\sin {\\tfrac {1}{2}}\\theta }}\\\\\[5pt\]\\sum \_{k=1}^{n}\\cos k\\theta &={\\frac {-\\sin {\\tfrac {1}{2}}\\theta +\\sin \\left(\\left(n+{\\tfrac {1}{2}}\\right)\\theta \\right)}{2\\sin {\\tfrac {1}{2}}\\theta }}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4679d7bb8112018d06e7d3a849bbe4ce024ff148) for 
A related function is the [Dirichlet kernel](https://en.wikipedia.org/wiki/Dirichlet_kernel "Dirichlet kernel"):
A similar identity is[\[41\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-41)
The proof is the following. By using the [angle sum and difference identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities),  Then let's examine the following formula,
 and this formula can be written by using the above identity,
So, dividing this formula with  completes the proof.
## Certain linear fractional transformations
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=31 "Edit section: Certain linear fractional transformations")\]
If  is given by the [linear fractional transformation](https://en.wikipedia.org/wiki/M%C3%B6bius_transformation "Möbius transformation")  and similarly  then 
More tersely stated, if for all  we let  be what we called  above, then 
If  is the slope of a line, then  is the slope of its rotation through an angle of 
## Relation to the complex exponential function
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=32 "Edit section: Relation to the complex exponential function")\]
Euler's formula states that, for any real number *x*:[\[42\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-42)  where *i* is the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit"). Substituting −*x* for *x* gives us: 
These two equations can be used to solve for cosine and sine in terms of the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function"). Specifically,[\[43\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-43)[\[44\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-44)  
These formulae are useful for proving many other trigonometric identities. For example, that *e**i*(*θ*\+*φ*) = *e**iθ* *e**iφ* means that
cos(*θ* + *φ*) + *i* sin(*θ* + *φ*) = (cos *θ* + *i* sin *θ*) (cos *φ* + *i* sin *φ*) = (cos *θ* cos *φ* − sin *θ* sin *φ*) + *i* (cos *θ* sin *φ* + sin *θ* cos *φ*).
That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.
The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the [complex logarithm](https://en.wikipedia.org/wiki/Complex_logarithm "Complex logarithm").
| Function | Inverse function[\[45\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-45) |
|---|---|
|  | |
## Relation to complex hyperbolic functions
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=33 "Edit section: Relation to complex hyperbolic functions")\]
Trigonometric functions may be deduced from [hyperbolic functions](https://en.wikipedia.org/wiki/Hyperbolic_functions "Hyperbolic functions") with [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") arguments. The formulae for the relations are shown below[\[46\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-46)[\[47\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-47).
When using a [power series](https://en.wikipedia.org/wiki/Power_series "Power series") expansion to define trigonometric functions, the following identities are obtained:[\[48\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-48)
## Infinite product formulae
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=35 "Edit section: Infinite product formulae")\]
For applications to [special functions](https://en.wikipedia.org/wiki/Special_functions "Special functions"), the following [infinite product](https://en.wikipedia.org/wiki/Infinite_product "Infinite product") formulae for trigonometric functions are useful:[\[49\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-49)[\[50\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-50)
![{\\displaystyle {\\begin{aligned}\\sin x&=x\\prod \_{n=1}^{\\infty }\\left(1-{\\frac {x^{2}}{\\pi ^{2}n^{2}}}\\right),&\\cos x&=\\prod \_{n=1}^{\\infty }\\left(1-{\\frac {x^{2}}{\\pi ^{2}\\left(n-{\\frac {1}{2}}\\right)\\!{\\vphantom {)}}^{2}}}\\right),\\\\\[10mu\]\\sinh x&=x\\prod \_{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{\\pi ^{2}n^{2}}}\\right),&\\cosh x&=\\prod \_{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{\\pi ^{2}\\left(n-{\\frac {1}{2}}\\right)\\!{\\vphantom {)}}^{2}}}\\right).\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d43e1176a38b1e6b653837eb782ae79735783ac)
## Inverse trigonometric functions
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=36 "Edit section: Inverse trigonometric functions")\]
The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[\[51\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-51)
Taking the [multiplicative inverse](https://en.wikipedia.org/wiki/Multiplicative_inverse "Multiplicative inverse") of both sides of the each equation above results in the equations for  The right hand side of the formula above will always be flipped. For example, the equation for  is:  while the equations for  and  are: 
The following identities are implied by the [reflection identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Reflections). They hold whenever  and  are in the domains of the relevant functions. ![{\\displaystyle {\\begin{alignedat}{9}{\\frac {\\pi }{2}}~&=~\\arcsin(x)&&+\\arccos(x)~&&=~\\arctan(r)&&+\\operatorname {arccot} (r)~&&=~\\operatorname {arcsec} (s)&&+\\operatorname {arccsc} (s)\\\\\[0.4ex\]\\pi ~&=~\\arccos(x)&&+\\arccos(-x)~&&=~\\operatorname {arccot} (r)&&+\\operatorname {arccot} (-r)~&&=~\\operatorname {arcsec} (s)&&+\\operatorname {arcsec} (-s)\\\\\[0.4ex\]0~&=~\\arcsin(x)&&+\\arcsin(-x)~&&=~\\arctan(r)&&+\\arctan(-r)~&&=~\\operatorname {arccsc} (s)&&+\\operatorname {arccsc} (-s)\\\\\[1.0ex\]\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/839db041f421b4af5890106450e2b4d3e9e183d4)
Also,[\[52\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Wu-52)   
The [arctangent](https://en.wikipedia.org/wiki/Arctangent "Arctangent") function can be expanded as a series:[\[53\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-53) 
## Identities without variables
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=37 "Edit section: Identities without variables")\]
In terms of the [arctangent](https://en.wikipedia.org/wiki/Arctangent "Arctangent") function we have[\[52\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Wu-52) 
The curious identity known as [Morrie's law](https://en.wikipedia.org/wiki/Morrie%27s_law "Morrie's law"), 
is a special case of an identity that contains one variable: 
Similarly,  is a special case of an identity with : 
For the case , 
For the case , 
The same cosine identity is 
Similarly, 
Similarly, 
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below): 
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators: 
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are [relatively prime](https://en.wikipedia.org/wiki/Coprime "Coprime") to (or have no [prime factors](https://en.wikipedia.org/wiki/Prime_factor "Prime factor") in common with) 21. The last several examples are corollaries of a basic fact about the irreducible [cyclotomic polynomials](https://en.wikipedia.org/wiki/Cyclotomic_polynomial "Cyclotomic polynomial"): the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the [Möbius function](https://en.wikipedia.org/wiki/M%C3%B6bius_function "Möbius function") evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.
Other cosine identities include:[\[54\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-54)  and so forth for all odd numbers, and hence 
Many of those curious identities stem from more general facts like the following:[\[55\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-55)  and 
Combining these gives us 
If n is an odd number () we can make use of the symmetries to get 
The transfer function of the [Butterworth low pass filter](https://en.wikipedia.org/wiki/Butterworth_filter "Butterworth filter") can be expressed in terms of polynomial and poles. By setting the frequency as the [cutoff frequency](https://en.wikipedia.org/wiki/Cutoff_frequency "Cutoff frequency"), the following identity can be proved: 
An efficient way to [compute π](https://en.wikipedia.org/wiki/Pi "Pi") to a [large number of digits](https://en.wikipedia.org/wiki/Approximations_of_pi "Approximations of pi") is based on the following identity without variables, due to [Machin](https://en.wikipedia.org/wiki/John_Machin "John Machin"). This is known as a [Machin-like formula](https://en.wikipedia.org/wiki/Machin-like_formula "Machin-like formula"):  or, alternatively, by using an identity of [Leonhard Euler](https://en.wikipedia.org/wiki/Leonhard_Euler "Leonhard Euler"):  or by using [Pythagorean triples](https://en.wikipedia.org/wiki/Pythagorean_triple "Pythagorean triple"): 
Others include:[\[56\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Harris-56)[\[52\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-Wu-52)   
Generally, for numbers *t*1, ..., *t**n*−1 ∈ (−1, 1) for which *θ**n* = Σ*n*−1
*k*\=1 arctan *t**k* ∈ (*π*/4, 3*π*/4), let *t**n* = tan(*π*/2 − *θ**n*) = cot *θ**n*. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are *t*1, ..., *t**n*−1 and its value will be in (−1, 1). In particular, the computed *t**n* will be rational whenever all the *t*1, ..., *t**n*−1 values are rational. With these values, 
where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the *t**k* values is not within (−1, 1). Note that if *t* = *p*/*q* is rational, then the (2*t*, 1 − *t*2, 1 + *t*2) values in the above formulae are proportional to the Pythagorean triple (2*pq*, *q*2 − *p*2, *q*2 + *p*2).
For example, for *n* = 3 terms,  for any *a*, *b*, *c*, *d* \> 0.
### An identity of Euclid
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=39 "Edit section: An identity of Euclid")\]
[Euclid](https://en.wikipedia.org/wiki/Euclid "Euclid") showed in Book XIII, Proposition 10 of his *[Elements](https://en.wikipedia.org/wiki/Euclid%27s_Elements "Euclid's Elements")* that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says: 
[Ptolemy](https://en.wikipedia.org/wiki/Ptolemy "Ptolemy") used this proposition to compute some angles in [his table of chords](https://en.wikipedia.org/wiki/Ptolemy%27s_table_of_chords "Ptolemy's table of chords") in Book I, chapter 11 of *[Almagest](https://en.wikipedia.org/wiki/Almagest "Almagest")*.
## Composition of trigonometric functions
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=40 "Edit section: Composition of trigonometric functions")\]
These identities involve a trigonometric function of a trigonometric function:[\[57\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-57)
where Ji are [Bessel functions](https://en.wikipedia.org/wiki/Bessel_function "Bessel function").
## Further "conditional" identities for the case *α* + *β* + *γ* = 180°
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=41 "Edit section: Further \"conditional\" identities for the case α + β + γ = 180°")\]
A **conditional trigonometric identity** is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied.[\[58\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-58) The following formulae apply to arbitrary plane triangles and follow from  as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur).[\[59\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-59) 
## Historical shorthands
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=42 "Edit section: Historical shorthands")\]
The [versine](https://en.wikipedia.org/wiki/Versine "Versine"), [coversine](https://en.wikipedia.org/wiki/Coversine "Coversine"), [haversine](https://en.wikipedia.org/wiki/Haversine "Haversine"), and [exsecant](https://en.wikipedia.org/wiki/Exsecant "Exsecant") were used in navigation. For example, the [haversine formula](https://en.wikipedia.org/wiki/Haversine_formula "Haversine formula") was used to calculate the distance between two points on a sphere. They are rarely used today.
The **[Dirichlet kernel](https://en.wikipedia.org/wiki/Dirichlet_kernel "Dirichlet kernel")** *Dn*(*x*) is the function occurring on both sides of the next identity: 
The [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution") of any [integrable function](https://en.wikipedia.org/wiki/Integrable_function "Integrable function") of period  with the Dirichlet kernel coincides with the function's th-degree Fourier approximation. The same holds for any [measure](https://en.wikipedia.org/wiki/Measure_\(mathematics\) "Measure (mathematics)") or [generalized function](https://en.wikipedia.org/wiki/Distribution_\(mathematics\) "Distribution (mathematics)").
### Tangent half-angle substitution
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=45 "Edit section: Tangent half-angle substitution")\]
If we set  then[\[60\]](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-60)  where  sometimes abbreviated to [cis](https://en.wikipedia.org/wiki/Cis_\(mathematics\) "Cis (mathematics)") *x*.
When this substitution of  for tan *x*/2 is used in [calculus](https://en.wikipedia.org/wiki/Calculus "Calculus"), it follows that  is replaced by 2*t*/1 + *t*2,  is replaced by 1 − *t*2/1 + *t*2 and the differential d*x* is replaced by 2 d*t*/1 + *t*2. Thereby one converts rational functions of  and  to rational functions of  in order to find their [antiderivatives](https://en.wikipedia.org/wiki/Antiderivative "Antiderivative").
### Viète's infinite product
\[[edit](https://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit§ion=46 "Edit section: Viète's infinite product")\]
- [Aristarchus's inequality](https://en.wikipedia.org/wiki/Aristarchus%27s_inequality "Aristarchus's inequality")
- [Derivatives of trigonometric functions](https://en.wikipedia.org/wiki/Table_of_derivatives#Derivatives_of_trigonometric_functions "Table of derivatives")
- [Exact trigonometric values](https://en.wikipedia.org/wiki/Exact_trigonometric_values "Exact trigonometric values") (values of sine and cosine expressed in surds)
- [Exsecant](https://en.wikipedia.org/wiki/Exsecant "Exsecant")
- [Half-side formula](https://en.wikipedia.org/wiki/Half-side_formula "Half-side formula")
- [Hyperbolic function](https://en.wikipedia.org/wiki/Hyperbolic_function "Hyperbolic function")
- Laws for solution of triangles:
- [Law of cosines](https://en.wikipedia.org/wiki/Law_of_cosines "Law of cosines")
- [Spherical law of cosines](https://en.wikipedia.org/wiki/Spherical_law_of_cosines "Spherical law of cosines")
- [Law of sines](https://en.wikipedia.org/wiki/Law_of_sines "Law of sines")
- [Law of tangents](https://en.wikipedia.org/wiki/Law_of_tangents "Law of tangents")
- [Law of cotangents](https://en.wikipedia.org/wiki/Law_of_cotangents "Law of cotangents")
- [Mollweide's formula](https://en.wikipedia.org/wiki/Mollweide%27s_formula "Mollweide's formula")
- [List of integrals of trigonometric functions](https://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions "List of integrals of trigonometric functions")
- [Mnemonics in trigonometry](https://en.wikipedia.org/wiki/Mnemonics_in_trigonometry "Mnemonics in trigonometry")
- [Pentagramma mirificum](https://en.wikipedia.org/wiki/Pentagramma_mirificum "Pentagramma mirificum")
- [Proofs of trigonometric identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities "Proofs of trigonometric identities")
- [Prosthaphaeresis](https://en.wikipedia.org/wiki/Prosthaphaeresis "Prosthaphaeresis")
- [Pythagorean theorem](https://en.wikipedia.org/wiki/Pythagorean_theorem "Pythagorean theorem")
- [Tangent half-angle formula](https://en.wikipedia.org/wiki/Tangent_half-angle_formula "Tangent half-angle formula")
- [Trigonometric number](https://en.wikipedia.org/wiki/Trigonometric_number "Trigonometric number")
- [Trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry")
- [Uses of trigonometry](https://en.wikipedia.org/wiki/Uses_of_trigonometry "Uses of trigonometry")
- [Versine](https://en.wikipedia.org/wiki/Versine "Versine") and [haversine](https://en.wikipedia.org/wiki/Haversine "Haversine")
1. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-AS4345_1-0)**
[Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene Ann](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun"), eds. (1983) \[June 1964\]. ["Chapter 4, eqn 4.3.45"](http://www.math.ubc.ca/~cbm/aands/page_73.htm). [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun"). Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 73. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-61272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-61272-0 "Special:BookSources/978-0-486-61272-0")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [64-60036](https://lccn.loc.gov/64-60036). [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0167642](https://mathscinet.ams.org/mathscinet-getitem?mr=0167642). [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [65-12253](https://www.loc.gov/item/65012253).
2. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-2)** [Selby 1970](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#CITEREFSelby1970), p. 188
3. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-3)** Abramowitz and Stegun, p. 72, 4.3.13–15
4. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-4)** Abramowitz and Stegun, p. 72, 4.3.7–9
5. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-5)** Abramowitz and Stegun, p. 72, 4.3.16
6. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_addition_6-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_addition_6-1) [***c***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_addition_6-2) [***d***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_addition_6-3)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Trigonometric Addition Formulas"](https://mathworld.wolfram.com/TrigonometricAdditionFormulas.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
7. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-7)** Abramowitz and Stegun, p. 72, 4.3.17
8. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-8)** Abramowitz and Stegun, p. 72, 4.3.18
9. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-:0_9-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-:0_9-1)
["Angle Sum and Difference Identities"](http://www.milefoot.com/math/trig/22anglesumidentities.htm). *www.milefoot.com*. Retrieved 2019-10-12.
10. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-10)** Abramowitz and Stegun, p. 72, 4.3.19
11. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-11)** Abramowitz and Stegun, p. 80, 4.4.32
12. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-12)** Abramowitz and Stegun, p. 80, 4.4.33
13. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-13)** Abramowitz and Stegun, p. 80, 4.4.34
14. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-14)**
Bronstein, Manuel (1989). "Simplification of real elementary functions". In Gonnet, G. H. (ed.). *Proceedings of the ACM-[SIGSAM](https://en.wikipedia.org/wiki/SIGSAM "SIGSAM") 1989 International Symposium on Symbolic and Algebraic Computation*. ISSAC '89 (Portland US-OR, 1989-07). New York: [ACM](https://en.wikipedia.org/wiki/Association_for_Computing_Machinery "Association for Computing Machinery"). pp. 207–211\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1145/74540.74566](https://doi.org/10.1145%2F74540.74566). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-89791-325-6](https://en.wikipedia.org/wiki/Special:BookSources/0-89791-325-6 "Special:BookSources/0-89791-325-6")
.
15. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-15)** Michael Hardy. (2016). "On Tangents and Secants of Infinite Sums." *The American Mathematical Monthly*, volume 123, number 7, 701–703. <https://doi.org/10.4169/amer.math.monthly.123.7.701>
16. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-16)**
Hardy, Michael (2025). "Invariance of the Cauchy Family Under Linear Fractional Transformations". *The American Mathematical Monthly*. **132** (5): 453–455\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/00029890.2025.2459048](https://doi.org/10.1080%2F00029890.2025.2459048).
17. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-17)**
Knight, F. B. (1976). "A characterization of the Cauchy type". *Proceedings of the American Mathematical Society*. **1976**: 130–135\.
18. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-18)**
Hardy, Michael (2016). ["On Tangents and Secants of Infinite Sums"](https://zenodo.org/record/1000408). *American Mathematical Monthly*. **123** (7): 701–703\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.4169/amer.math.monthly.123.7.701](https://doi.org/10.4169%2Famer.math.monthly.123.7.701).
19. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-cut-the-knot.org_19-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-cut-the-knot.org_19-1)
["Sine, Cosine, and Ptolemy's Theorem"](https://www.cut-the-knot.org/proofs/sine_cosine.shtml).
20. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_multiple_angle_20-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_multiple_angle_20-1)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Multiple-Angle Formulas"](https://mathworld.wolfram.com/Multiple-AngleFormulas.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
21. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-21)** Abramowitz and Stegun, p. 74, 4.3.48
22. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-STM1_22-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-STM1_22-1) [Selby 1970](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#CITEREFSelby1970), pg. 190
23. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-A&S_23-0)** Cite error: The named reference `A&S` was invoked but never defined (see the [help page](https://en.wikipedia.org/wiki/Help:Cite_errors/Cite_error_references_no_text "Help:Cite errors/Cite error references no text")).
24. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-24)**
Weisstein, Eric W. ["Multiple-Angle Formulas"](https://mathworld.wolfram.com/Multiple-AngleFormulas.html). *mathworld.wolfram.com*. Retrieved 2022-02-06.
25. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-25)**
Ward, Ken. ["Multiple angles recursive formula"](http://trans4mind.com/personal_development/mathematics/trigonometry/multipleAnglesRecursiveFormula.htm). *Ken Ward's Mathematics Pages*.
26. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-ReferenceA_26-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-ReferenceA_26-1)
[Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene Ann](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun"), eds. (1983) \[June 1964\]. ["Chapter 4, eqn 4.3.20-22"](http://www.math.ubc.ca/~cbm/aands/page_72.htm). [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun"). Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 72. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-61272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-61272-0 "Special:BookSources/978-0-486-61272-0")
. [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [64-60036](https://lccn.loc.gov/64-60036). [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [0167642](https://mathscinet.ams.org/mathscinet-getitem?mr=0167642). [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [65-12253](https://www.loc.gov/item/65012253).
27. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_half_angle_27-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_half_angle_27-1)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Half-Angle Formulas"](https://mathworld.wolfram.com/Half-AngleFormulas.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
28. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-28)** Abramowitz and Stegun, p. 72, 4.3.24–26
29. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-mathworld_double_angle_29-0)**
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Double-Angle Formulas"](https://mathworld.wolfram.com/Double-AngleFormulas.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
30. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Stegun_p._72,_4_30-0)** Abramowitz and Stegun, p. 72, 4.3.27–28
31. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-31)** Abramowitz and Stegun, p. 72, 4.3.31–33
32. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-32)**
Eves, Howard (1990). *An introduction to the history of mathematics* (6th ed.). Philadelphia: Saunders College Pub. p. 309. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-03-029558-0](https://en.wikipedia.org/wiki/Special:BookSources/0-03-029558-0 "Special:BookSources/0-03-029558-0")
. [OCLC](https://en.wikipedia.org/wiki/OCLC_\(identifier\) "OCLC (identifier)") [20842510](https://search.worldcat.org/oclc/20842510).
33. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-A&S_sum-to-product_33-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-A&S_sum-to-product_33-1) Abramowitz and Stegun, p. 72, 4.3.34–39
34. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-34)**
Johnson, Warren P. (Apr 2010). "Trigonometric Identities à la Hermite". *[American Mathematical Monthly](https://en.wikipedia.org/wiki/American_Mathematical_Monthly "American Mathematical Monthly")*. **117** (4): 311–327\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.4169/000298910x480784](https://doi.org/10.4169%2F000298910x480784). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [29690311](https://api.semanticscholar.org/CorpusID:29690311).
35. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-35)**
["Product Identity Multiple Angle"](https://math.stackexchange.com/q/2095330).
36. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-36)** Apostol, T.M. (1967) Calculus. 2nd edition. New York, NY, Wiley. Pp 334-335.
37. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-ReferenceB_37-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-ReferenceB_37-1)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Harmonic Addition Theorem"](https://mathworld.wolfram.com/HarmonicAdditionTheorem.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
38. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Muniz_38-0)**
Ortiz Muñiz, Eddie (Feb 1953). "A Method for Deriving Various Formulas in Electrostatics and Electromagnetism Using Lagrange's Trigonometric Identities". *American Journal of Physics*. **21** (2): 140. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1953AmJPh..21..140M](https://ui.adsabs.harvard.edu/abs/1953AmJPh..21..140M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.1933371](https://doi.org/10.1119%2F1.1933371).
39. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-39)**
Agarwal, Ravi P.; O'Regan, Donal (2008). [*Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems*](https://books.google.com/books?id=jWvAfcNnphIC) (illustrated ed.). Springer Science & Business Media. p. 185. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-79146-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-79146-3 "Special:BookSources/978-0-387-79146-3")
.
[Extract of page 185](https://books.google.com/books?id=jWvAfcNnphIC&pg=PA185)
40. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-40)**
Jeffrey, Alan; Dai, Hui-hui (2008). "Section 2.4.1.6". *Handbook of Mathematical Formulas and Integrals* (4th ed.). Academic Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-12-374288-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-374288-9 "Special:BookSources/978-0-12-374288-9")
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41. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-41)**
Fay, Temple H.; Kloppers, P. Hendrik (2001). ["The Gibbs' phenomenon"](https://dx.doi.org/10.1080/00207390117151). *International Journal of Mathematical Education in Science and Technology*. **32** (1): 73–89\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/00207390117151](https://doi.org/10.1080%2F00207390117151).
42. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-42)** Abramowitz and Stegun, p. 74, 4.3.47
43. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-43)** Abramowitz and Stegun, p. 71, 4.3.2
44. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-44)** Abramowitz and Stegun, p. 71, 4.3.1
45. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-45)** Abramowitz and Stegun, p. 80, 4.4.26–31
46. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-46)**
Hawkins, Faith Mary; Hawkins, J. Q. (March 1, 1969). [*Complex Numbers and Elementary Complex Functions*](https://archive.org/details/isbn_356025055/mode/2up). London: MacDonald Technical & Scientific London (published 1968). p. 122. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0356025056](https://en.wikipedia.org/wiki/Special:BookSources/978-0356025056 "Special:BookSources/978-0356025056")
.
47. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-47)**
Markushevich, A. I. (1966). [*The Remarkable Sine Function*](https://archive.org/details/markushevich-the-remarkable-sine-functions). New York: American Elsevier Publishing Company, Inc. pp. 35–37, 81. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1483256313](https://en.wikipedia.org/wiki/Special:BookSources/978-1483256313 "Special:BookSources/978-1483256313")
.
48. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-48)** Abramowitz and Stegun, p. 74, 4.3.65–66
49. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-49)** Abramowitz and Stegun, p. 75, 4.3.89–90
50. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-50)** Abramowitz and Stegun, p. 85, 4.5.68–69
51. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-51)** [Abramowitz & Stegun 1972](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#CITEREFAbramowitzStegun1972), p. 73, 4.3.45
52. ^ [***a***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Wu_52-0) [***b***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Wu_52-1) [***c***](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Wu_52-2) Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", *Mathematics Magazine* 77(3), June 2004, p. 189.
53. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-53)**
S. M. Abrarov; R. K. Jagpal; R. Siddiqui; B. M. Quine (2021), "Algorithmic determination of a large integer in the two-term Machin-like formula for π", *Mathematics*, **9** (17), 2162, [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[2107\.01027](https://arxiv.org/abs/2107.01027), [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.3390/math9172162](https://doi.org/10.3390%2Fmath9172162)
54. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-54)**
Humble, Steve (Nov 2004). "Grandma's identity". *Mathematical Gazette*. **88**: 524–525\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/s0025557200176223](https://doi.org/10.1017%2Fs0025557200176223). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [125105552](https://api.semanticscholar.org/CorpusID:125105552).
55. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-55)**
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Sine"](https://mathworld.wolfram.com/Sine.html). *[MathWorld](https://en.wikipedia.org/wiki/MathWorld "MathWorld")*.
56. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-Harris_56-0)** Harris, Edward M. "Sums of Arctangents", in Roger B. Nelson, *Proofs Without Words* (1993, Mathematical Association of America), p. 39.
57. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-57)** [Milton Abramowitz and Irene Stegun, *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun"), [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"), New York, 1972, formulae 9.1.42–9.1.45
58. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-58)** Er. K. C. Joshi, *Krishna's IIT MATHEMATIKA*. Krishna Prakashan Media. Meerut, India. page 636.
59. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-59)** Cagnoli, Antonio (1808), *Trigonométrie rectiligne et sphérique*, p. 27.
60. **[^](https://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_ref-60)** Abramowitz and Stegun, p. 72, 4.3.23
- [Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene A.](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun"), eds. (1972). [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://archive.org/details/handbookofmathe000abra). New York: [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-61272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-61272-0 "Special:BookSources/978-0-486-61272-0")
.
- Nielsen, Kaj L. (1966), *Logarithmic and Trigonometric Tables to Five Places* (2nd ed.), New York: [Barnes & Noble](https://en.wikipedia.org/wiki/Barnes_%26_Noble "Barnes & Noble"), [LCCN](https://en.wikipedia.org/wiki/LCCN_\(identifier\) "LCCN (identifier)") [61-9103](https://lccn.loc.gov/61-9103)
- Selby, Samuel M., ed. (1970), *Standard Mathematical Tables* (18th ed.), The Chemical Rubber Co.
- [Values of sin and cos, expressed in surds, for integer multiples of 3° and of 5\+5/8°](http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html), and for the same angles [csc and sec](http://www.jdawiseman.com/papers/easymath/surds_csc_sec.html) and [tan](http://www.jdawiseman.com/papers/easymath/surds_tan.html) |
| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/List_of_trigonometric_identities s443 |