Trigonometry / Unit 6: Trigonometric Identities
6.2 Simplifying Trig Expressions
Notes
To simplify, rewrite in terms of sine and cosine, factor, combine fractions, and use Pythagorean identities.
- Example: $\displaystyle \tan x \cos x = \frac{\sin x}{\cos x} \cdot \cos x = \sin x$.
- Look for a difference of squares: $(1 - \cos x)(1 + \cos x) = 1 - \cos^2 x = \sin^2 x$.
- Look for a common factor: $\sec x - \sec x \sin^2 x = \sec x (1 - \sin^2 x) = \sec x \cos^2 x = \cos x$.
Practice
Simplify $\tan x \cos x$.
$\displaystyle \frac{\sin x}{\cos x} \cdot \cos x = \sin x$.
Simplify $(1 - \cos x)(1 + \cos x)$.
Difference of squares gives $1 - \cos^2 x = \sin^2 x$.
Simplify $\sec x - \sec x \sin^2 x$.
Factor: $\sec x(1 - \sin^2 x) = \sec x \cos^2 x = \cos x$.
Ready for more? Take the Unit 6 test.