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 203: 57

Answer

$$\frac{\sec^4\theta-\tan^4\theta}{\sec^2\theta+\tan^2\theta}=\sec^2\theta-\tan^2\theta$$ We simplify the left side and find that the expression is an identity.

Work Step by Step

$$\frac{\sec^4\theta-\tan^4\theta}{\sec^2\theta+\tan^2\theta}=\sec^2\theta-\tan^2\theta$$ The left side is more complicated. We would simplify it. $$A=\frac{\sec^4\theta-\tan^4\theta}{\sec^2\theta+\tan^2\theta}$$ We have $a^4-b^4=(a^2-b^2)(a^2+b^2)$. So, $$A=\frac{(\sec^2\theta-\tan^2\theta)(\sec^2\theta+\tan^2\theta)}{\sec^2\theta+\tan^2\theta}$$ $$A=\sec^2\theta-\tan^2\theta$$ They are thus equal. The expression is an identity.
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