Verifying Trigonometric Identities

By the end of this lesson, you will be able to verify trig identities by transforming one side into the other using algebraic manipulation and known identities, and apply systematic strategies for complex verifications.

The Big Idea

Verifying a trig identity means proving it is true for all values in its domain — not solving for specific values. The key constraint: you transform one side (usually the more complicated side) into the other without "crossing the equal sign" or using operations that assume the result is already true.

Rules for Verifying

1. Work with one side only (usually the more complex)
2. Use known identities (reciprocal, quotient, Pythagorean, sum/difference, double-angle)
3. Don't add the same thing to both sides or cross-multiply
4. If stuck, convert to sines and cosines — the universal strategy

Strategies

1. Convert to sines and cosines — most reliable approach
2. Factor — look for difference of squares, common factors
3. Multiply by conjugate — to rationalize or create Pythagorean identity
4. Split fractions or combine fractions
5. Substitute a Pythagorean identity — create a form you recognize

Example 1 — Basic Verification

Verify: $\dfrac{\sin\theta}{\tan\theta} = \cos\theta$

Left side: $\dfrac{\sin\theta}{\sin\theta/\cos\theta} = \sin\theta \cdot \dfrac{\cos\theta}{\sin\theta} = \cos\theta$ ✓

Example 2 — Using Pythagorean Identity

Verify: $\dfrac{\sec^2\theta - 1}{\sec^2\theta} = \sin^2\theta$

Left: $\dfrac{\tan^2\theta}{\sec^2\theta} = \dfrac{\sin^2\theta/\cos^2\theta}{1/\cos^2\theta} = \sin^2\theta$ ✓

(Used $\sec^2\theta - 1 = \tan^2\theta$)

Example 3 — Convert Everything to sin/cos

Verify: $\sec\theta + \csc\theta = \dfrac{\sin\theta + \cos\theta}{\sin\theta\cos\theta}$

Right side: $\dfrac{\sin\theta + \cos\theta}{\sin\theta\cos\theta} = \dfrac{\sin\theta}{\sin\theta\cos\theta} + \dfrac{\cos\theta}{\sin\theta\cos\theta} = \dfrac{1}{\cos\theta} + \dfrac{1}{\sin\theta} = \sec\theta + \csc\theta$ ✓