Trigonometric identities

An identity is an equation that is true for every value of the variable. Trigonometric identities let you rewrite an expression in a simpler or more useful form. They are used constantly in calculus and physics.

This chapter starts with the basic identities from Chapter 1 and shows how to prove new ones. It then builds formulas for sums, differences, double angles, and half angles.

This chapter covers the following topics:

Proving identities

The fundamental identities, simplifying trigonometric expressions, and step-by-step ways to prove that an identity is true.

Sum and difference formulas

Formulas for the sine, cosine, and tangent of ${A+B}$ and ${A-B}$, exact values such as ${\cos 75^\circ}$, and writing ${a\sin x+b\cos x}$ as one sine wave.

Double-angle and half-angle formulas

Formulas for ${\sin 2x}$, ${\cos 2x}$, and ${\tan 2x}$, formulas for lowering powers, and formulas for half an angle.

Product-to-sum and sum-to-product formulas

Writing a product of sines and cosines as a sum, and a sum as a product, with an application to beats in sound.

Practice questions

Practice problems on every section of this chapter, with full step-by-step solutions.