## Precalculus (6th Edition)

Published by Pearson

# Chapter 7 - Trigonometric Identities and Equations - 7.2 Verifying Trigonometric Identities - 7.2 Exercises - Page 667: 53

#### Answer

$\frac{\cos\alpha}{\sec\alpha}+\frac{\sin\alpha}{\csc\alpha}=\sec^2\alpha-\tan^2\alpha$

#### Work Step by Step

Simplify the left side: $\frac{\cos\alpha}{\sec\alpha}+\frac{\sin\alpha}{\csc\alpha}$ $=\frac{\cos\alpha}{\frac{1}{\cos\alpha}}+\frac{\sin\alpha}{\frac{1}{\sin\alpha}}$ $=\frac{\cos\alpha}{\frac{1}{\cos\alpha}}*\frac{\cos\alpha}{\cos\alpha}+\frac{\sin\alpha}{\frac{1}{\sin\alpha}}\frac{\sin\alpha}{\sin\alpha}$ $=\cos^2\alpha+\sin^2\alpha$ $=1$ Simplify the right side: $\sec^2\alpha-\tan^2\alpha$ $=1$ Since both sides are equal to $1$, they are equal to each other, and the identity is proven.

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