Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 7 - Analytic Trigonometry - 7.4 Trigonometric Identities - 7.4 Assess Your Understanding - Page 476: 29


Work on the left side of the identity using $\sec{\theta}=\frac{1}{\cos\theta}$ and $1-\sin^\theta=\cos^2\theta$. Refer to the step-by-step part below for the complete proof.

Work Step by Step

We have to show that: $(\sec\theta+1)(\sec\theta-1)=\tan^2\theta$ By evaluating the left side we get: $=\sec^2\theta+\sec\theta-\sec\theta-1\\ =\sec^2\theta-1$ Since $\sec\theta=\frac{1}{\cos\theta}$ and $1-\cos^2\theta=\sin^2\theta$, the expression above simplifies to: $=\dfrac{1}{\cos^2\theta}-\dfrac{\cos^2\theta}{\cos^2\theta}\\ =\dfrac{1-\cos^2\theta}{\cos^2\theta}\\ =\dfrac{\sin^2\theta}{\cos^2\theta}\\ =\tan^2\theta$ Since LHS=RHS, the proof is complete
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