# Introduction to Trigonometric Identities - Video Tutorials & Practice Problems

## Even and Odd Identities

## Example 1

Use the even-odd identities to evaluate the expression.

$\cos\left(-\theta\right)-\cos\theta$

0

$-\cos\theta$

$2\cos\theta$

$-2\cos\theta$

Use the even-odd identities to evaluate the expression.

$-\cot\left(\theta\right)\cdot\sin\left(-\theta\right)$

$\tan\theta$

$-\cos\theta$

$\cos\theta$

$\frac{\cos\theta}{\sin^2\theta}$

Select the expression with the same value as the given expression.

$\sec\left(-\frac{4\pi}{5}\right)$

$\cos\left(\frac{4\pi}{5}\right)$

$-\cos\left(\frac{4\pi}{5}\right)$

$\sec\left(\frac{4\pi}{5}\right)$

$-\sec\left(\frac{4\pi}{5}\right)$

Select the expression with the same value as the given expression.

$\sin\left(-38\degree\right)$

$\sin38\degree$

$-\sin38\degree$

$-\sin\left(-38\degree\right)$

$\frac{1}{-\sin38\degree}$

## Pythagorean Identities

## Example 2

Use the Pythagorean identities to rewrite the expression as a single term.

$\left(1+\csc\theta\right)\left(1-\csc\theta\right)$

1

$-\csc^2\theta$

$\cot^2\theta$

$-\cot^2\theta$

Use the Pythagorean identities to rewrite the expression with no fraction.

$\frac{1}{1-\sec\theta}$

$1+\sec\theta$

$\frac{1}{\tan^2\theta}$

$-\cot^2\theta\left(1+\sec\theta\right)$

$-\tan^2\theta\left(1+\sec\theta\right)$

## Simplifying Trig Expressions

## Example 3

## Example 4

Simplify the expression.

$\tan^2\theta-\sec^2\theta+1$

0

1

$\csc^2\theta+1$

2

Simplify the expression.

$\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}$

$\sin\theta$

$-\sin\theta$

$-\cot\theta$

1

Simplify the expression.

$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)\csc^2\left(\theta\right)\cos^2\left(-\theta\right)$

$\cot^2\theta$

$\tan\theta$

1

– 1

## Verifying Trig Equations as Identities

## Example 5

## Example 6

Identify the most helpful first step in verifying the identity.

$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)=\sec^2\theta\sin^2\left(-\theta\right)$

Add the terms on the left side using a common denominator.

Rewrite left side of equation in terms of sine and cosine.

Use even-odd identity to eliminate negative argument on right side of equation.

Rewrite right side of equation in terms of sine and cosine.

Identify the most helpful first step in verifying the identity.

$\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}$

Rewrite left side of equation in terms of sine and cosine.

Subtract $\sec\theta$ from both sides.

Use reciprocal identity to rewrite $\sec\theta$ on right side of equation.

Rewrite $\tan^2\theta$ in terms of sine and cosine.