Integrated Math 3 / Unit 6: Trigonometric Identities and Laws
6.2 Simplifying and Verifying Identities
Notes
An identity is true for every value where both sides are defined. Useful identities:
- $\displaystyle \tan x = \frac{\sin x}{\cos x}$, $\displaystyle \cot x = \frac{\cos x}{\sin x}$, $\displaystyle \sec x = \frac{1}{\cos x}$, $\displaystyle \csc x = \frac{1}{\sin x}$.
- $\sin^2 x + \cos^2 x = 1$.
- To verify an identity, work on one side only. Rewrite in terms of sine and cosine, then simplify.
- Example: $\displaystyle \sec x - \cos x = \frac{1 - \cos^2 x}{\cos x} = \frac{\sin^2 x}{\cos x} = \sin x \tan x$.
- To show something is not an identity, find one counterexample: $\sin x + \cos x = 1$ fails at $\displaystyle x = \frac{\pi}{4}$.
Practice
Simplify $\tan x \cos x$.
$\displaystyle \frac{\sin x}{\cos x} \cdot \cos x = \sin x$.
Simplify $\displaystyle \frac{1 - \cos^2 x}{\sin x}$.
$1 - \cos^2 x = \sin^2 x$, so the fraction is $\displaystyle \frac{\sin^2 x}{\sin x} = \sin x$.
Which equation is NOT an identity?
At $\displaystyle x = \frac{\pi}{4}$, $\displaystyle \sin x + \cos x = \sqrt{2} \approx 1.41$, not 1.
Ready for more? Take the Unit 6 test.