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}.

### From the 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 in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivre's formula with .

### 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.

### 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 trig identity.

### 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

### Identities with no variables

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

### 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 that

- (verified using the unit circle and squeeze theorem or L'Hôpital's rule)

### 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*.