Precalculus / Unit 5: Trigonometric Identities and Equations
5.2 Simplifying and Verifying Identities
Notes
To verify an identity, work on one side only until it matches the other.
- Rewrite everything in terms of $\sin$ and $\cos$.
- Factor, combine fractions over a common denominator, or use a Pythagorean identity.
- Never move terms across the equals sign; do not treat the identity as an equation to solve.
- Example: $\displaystyle \sec x - \cos x = \frac{1}{\cos x} - \cos x = \frac{1 - \cos^2 x}{\cos x} = \frac{\sin^2 x}{\cos x} = \sin x \tan x$.
Practice
Simplify $\tan x \cos x$.
$\displaystyle \frac{\sin x}{\cos x} \cdot \cos x = \sin x$.
Simplify $(1 + \sin x)(1 - \sin x)$.
Difference of squares: $1 - \sin^2 x = \cos^2 x$.
Simplify $\csc x \tan x$.
$\displaystyle \frac{1}{\sin x} \cdot \frac{\sin x}{\cos x} = \frac{1}{\cos x}$.
Ready for more? Take the Unit 5 test.