## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 5 - Trigonometric Identities - Section 5.1 Fundamental Identities - 5.1 Exercises - Page 201: 40

#### Answer

$$\sec^2 x-1=\frac{\sin^2 x}{\cos^2 x}$$ $\text{A}$ is the answer.

#### Work Step by Step

$$A=\sec^2 x-1$$ A Pythagorean Identity related to $\sec^2\theta$ states that $$\sec^2\theta=\tan^2\theta+1$$ That means we can rewrite $A$ as follows: $$A=\tan^2 x+1-1$$ $$A=\tan^2 x$$ Also, from Quotient Identities, we know $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$ Therefore, $$A=\Bigg(\frac{\sin x}{\cos x}\Bigg)^2$$ $$A=\frac{\sin^2 x}{\cos^2 x}$$ $\text{A}$ is the answer.

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