List of trigonometric identities

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

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 :

in terms of

Reflections, shifts, and periodicity

By examining the unit circle, one can establish the following properties of the trigonometric functions.

Reflections

When the direction of a Euclidean vector is represented by an angle this is the angle determined by the free vector 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 with direction is reflected about a line with direction then the direction angle of this reflected line 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.

reflected in odd/even identities	reflected in	reflected in	reflected in	reflected in compare to

Shifts and periodicity

Shift by one quarter period	Shift by one half period	Shift by full periods	Period

Signs

The sign of trigonometric functions depends on quadrant of the angle. If and is the sign function,
The trigonometric functions are periodic with common period so for values of outside the interval they take repeating values.
The sign of a sinusoid or cosinusoid can be used to define a normalized square wave.
For example, the functions and take values and correspond to square waves with a phase shift of.

Angle sum and difference identities

These are also known as the .
The angle difference identities for and can be derived from the angle sum versions 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.

Sines and cosines of sums of infinitely many angles

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

Let be the 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. The case of infinitely many terms can be proved by using some elementary inequalities.

Linear fractional transformations of tangents, related to tangents of sums

Suppose and and
and let be any number for which
Suppose that so that the forgoing fraction cannot be. Then for all
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.

Secants and cosecants of sums

where is the th-degree elementary symmetric polynomial in the 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. The case of only finitely many terms can be proved by mathematical induction on the number of such terms.
For example,

Ptolemy's theorem

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. 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.

Multiple-angle and half-angle formulae

Multiple-angle formulae

Double-angle formulae

Formulae for twice an angle.

Triple-angle formulae

Formulae for triple angles.

Multiple-angle formulae

Formulae for multiple angles.

Chebyshev method

The Chebyshev method is a recursive algorithm for finding the th multiple angle formula knowing the th and th values.
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 formulae

Also

Table

These can be shown by using either the sum and difference identities or the multiple-angle formulae.

	Sine	Cosine	Tangent	Cotangent
Double-angle formula
Triple-angle formula
Half-angle formula

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, where is the value of the cosine function at the one-third angle and 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. None of these solutions are reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.