10.6 Proving Trigonometric Identities

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What Is a Trigonometric Identity?

A trigonometric identity is an equation involving trigonometric functions that is always true — no matter what value of the angle you substitute in.

For example, you already know that:

$\sin^2 x + \cos^2 x = 1$

This is true for every value of $x$. That makes it an identity.

Proving an identity means showing, using logical algebraic steps, that the left-hand side (LHS) and the right-hand side (RHS) of the equation are exactly the same expression.

The Six Trigonometric Functions — A Quick Reminder

Before proving identities, you need to know all six trig functions and how they connect. These come from Topic 10.4:

Identity Group 1 — The Pythagorean Identities

$\boxed{\sin^2 x + \cos^2 x = 1}$

By dividing every term in the above by $\cos^2 x$:

$\boxed{1 + \tan^2 x = \sec^2 x}$

By dividing every term in the first identity by $\sin^2 x$:

$\boxed{1 + \cot^2 x = \cosec^2 x}$

Identity Group 2 — The Ratio/Reciprocal Identities

$\tan x = \frac{\sin x}{\cos x} \qquad \cot x = \frac{\cos x}{\sin x}$

$\cosec x = \frac{1}{\sin x} \qquad \sec x = \frac{1}{\cos x} \qquad \cot x = \frac{1}{\tan x}$