# Trigonometric identity

In mathematics,**trigonometric identities**are equalities involving trigonometric functions that are true for all values of the occurring variables. 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 trick involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

**Notation**: With trigonometric functions, we define functions sin^{2}, cos^{2}, etc., such that sin^{2}(*x*) = (sin(*x*))^{2}. Often, sin^{−1}(*x*) is used to denote the inverse function. In this article, we prefer to write either arcsin(*x*) to indicate the inverse function, or csc(*x*) to indicate the multiplicative inverse.

## Definitions

## Periodicity, symmetry and shifts

These are most easily shown from the unit circle:

## From the Pythagorean theorem

## Addition/subtraction theorems

The quickest way to prove these is Euler's formula. The tangent formula follows from the other two. A geometric proof of the sin(*x* + *y*) identity is given at the end of this article.

## Double-angle formulas

These can be shown by substituting *x* = *y* in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivre's formula with *n* = 2.

## Multiple-angle formulas

If *T _{n}* is the

*n*th Chebyshev polynomial then

**Dirichlet kernel**

*D*(

_{n}*x*) is the function occurring on both sides of the next identity:

*n*th-degree Fourier approximation. The same holds for any measure or generalized function.

## Power-reduction formulas

Solve the third and fourth double angle formula for cos^{2}(*x*) and sin^{2}(*x*).

## Half-angle formulas

Substitute *x*/2 for *x* in the power reduction formulas, then solve for cos(*x*/2) and sin(*x*/2).

*x*/2) by 2cos(

*x*/2) / ( 2cos(

*x*/2)) and substitute sin(

*x*/2) / cos(

*x*/2) for tan(

*x*/2). The numerator is then sin(

*x*) via the double-angle formula, and the denominator is 2cos

^{2}(

*x*/2) − 1 + 1, which is cos(

*x*) + 1 by the double-angle formulae. The second formula comes from the first formula multiplied by sin(

*x*) / sin(

*x*) and simplified using the Pythagorean trigonometric identity.

and | and |

This substitution of *t* for tan(*x*/2), with the consequent replacement of sin(*x*) by 2*t*/(1 + *t*^{2}) and cos(*x*) by (1 − *t*^{2})/(1 + *t*^{2}) is useful in calculus for converting rational functions in in sin(*x*) and cos(*x*) to functions of *t* in order to find their antiderivatives. (See "abstract point of view" below.)

## Products to sums

These can be proven by expanding their right-hand-sides using the addition theorems.

## Sums to products

Replace *x* by (*x* + *y*) / 2 and *y* by (*x* – *y*) / 2 in the Product-to-Sum formulas.

## Inverse trigonometric functions

## The Gudermannian function

The Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers -- see that article for details.

## Identities with no variables

Richard Feynman is reputed to have learned as a boy, and always remembered, the following curious identity:

The following identity with no variables can be used to compute π efficiently:

## Calculus

In calculus it is essential that angles that are arguments to trigonometric functions be measured in radians; if they are measured in degrees or any other units, then the relations stated below fail. If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by first verifying two limits.

- (verified using the identity tan(
*x*/2) = (1 − cos(*x*))/sin(*x*))

## Proofs using a differential equation

Consider this differential equation:

Using Euler's formula and the method for solving linear differential equations combined with the uniqueness theorem and the existence theorem we can define sine and cosine as the following1

is the unique solution of

is the unique solution of

Now, let's prove that

*T(x)*

- but since is a solution of we can say so

*B*by plugging in 0 for

*x*

*A*we take the derviative of

*T(x)*and plug in 0 for

*x*

*A*and

*B*back into our original equation for

*T(x)*we get

*T(x)*was defined as we get

Using these rigorous definitions of sine and cosine, you can prove all the other properties of sine and cosine using the same technique.

## Geometric proofs

### sin(*x* + *y*) sin(*x*) cos(*y*) + cos(*x*) sin(*y*)

Angle *x* = Angle BAC = Angle ACE = Angle CDE.

### cos(*x* + *y*) cos(*x*) cos(*y*) − sin(*x*) sin(*y*)

Using the above figure:

;

## Abstract point of view

Since the circle is an algebraic curve of genus 0, one expects the 'circular functions' to be reducible to rational functions. This is known classically, by systematically using the *tan-half-angle* formulae to write the sine and cosine functions in terms of a new variable *t*.