Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 202: 39

Answer

$$\cos^2\theta(\tan^2\theta+1)=1$$ We simplify the left side first. It is proved to be an identity.

Work Step by Step

$$\cos^2\theta(\tan^2\theta+1)=1$$ We simplify the left side, since it is obviously more complex. $$A=\cos^2\theta(\tan^2\theta+1)$$ We can change $\tan^2\theta+1$ according to the following Pythagorean Identity: $$\tan^2\theta+1=\sec^2\theta$$ Also from a Reciprocal Identity, $$\sec\theta=\frac{1}{\cos\theta}$$ That means $$\sec^2\theta=\frac{1}{\cos^2\theta}$$ Therefore, $$\tan^2\theta+1=\frac{1}{\cos^2\theta}$$ We now apply it into $A$: $$A=\cos^2\theta\times\frac{1}{\cos^2\theta}$$ $$A=1$$ The left side and right side are equal. The trigonometric expression is an identity.
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