## Precalculus (6th Edition)

Published by Pearson

# Chapter 7 - Trigonometric Identities and Equations - 7.2 Verifying Trigonometric Identities - 7.2 Exercises - Page 667: 51

#### Answer

$\cot\theta+\tan\theta=\sec\theta\csc\theta$

#### Work Step by Step

Start with the left side: $\cot\theta+\tan\theta$ Rewrite in terms of sine and cosine: $=\frac{\cos\theta}{\sin\theta}+\frac{\sin\theta}{\cos\theta}$ Get a common denominator: $=\frac{\cos\theta}{\sin\theta}*\frac{\cos\theta}{\cos\theta}+\frac{\sin\theta}{\cos\theta}*\frac{\sin\theta}{\sin\theta}$ $=\frac{\cos^2\theta}{\sin\theta\cos\theta}+\frac{\sin^2\theta}{\sin\theta\cos\theta}$ $=\frac{\cos^2\theta+\sin^2\theta}{\sin\theta\cos\theta}$ Simplify: $=\frac{1}{\sin\theta\cos\theta}$ $=\frac{1}{\cos\theta}*\frac{1}{\sin\theta}$ $=\sec\theta\csc\theta$ Since this equals the right side, the identity has been proven.

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