Section 9.1 - Basic Trigonometric Identities

Section Objectives

1. Use basic trigonometric identities to simplify and evaluate trigonometric expressions.
2. Verify (prove) trigonometric identities.

Using Basic Identities to Simplify

Basic identities are extremely useful for simplifying expressions. Here are some examples....

Examples

• Suppose $\sec u = -3/2$ and $\tan u >0$. Find the exact values of all six trigonometric functions at $u$. (We could use a right triangle, but let's use a Pythagorean identity.)

• Simplify: $\quad \cos^2 x \csc x - \csc x$

• Rewrite and factor: $\quad \sec^2 x + 3\tan x + 1$

• Simplify: $\quad \displaystyle \frac{\sin \theta}{1+\cos \theta} + \frac{\cos \theta}{\sin \theta}$

• Use the substitution $x=2 \tan \theta$, $0 < \theta < \pi/2$, to write $\sqrt{x^2+4}$ in terms of $\theta$.

Verifying Identities

Verifying trig identities can be very difficult---it's hard to figure out where to start, and it's easy to get stuck in cycles. Here is some advice:

1. Always remember that there is no single correct approach.
2. Work on one side at a time.
3. Look for opportunities to factor, get common denominators and add fractions, expand algebraic expressions, multiply by conjugates, etc.
4. Be on the lookout for opportunities to use fundamental identities.
5. If all else fails, consider converting to sines and cosines.
6. Do something! Quite often, the action of just trying something will lead to useful results.

Examples

• Verify: $\quad \displaystyle \frac{\sin^2 \theta + \cos^2 \theta}{\cos^2 \theta \, \sec^2 \theta}=1$

• Verify: $\quad \displaystyle 2 \sec^2 \beta = \frac{1}{1- \sin \beta} + \frac{1}{1 + \sin \beta}$

• Verify: $\quad (\tan^2 x + 1)(\cos^2 x -1) = -\tan^2 x$

• Verify: $\quad \displaystyle \csc \alpha + \cot \alpha = \frac{\sin \alpha}{1-\cos \alpha}$

• Verify: $\quad \displaystyle \frac{\cot^2 x}{1+\csc x} = \frac{1- \sin x}{\sin x}$