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: 36

Answer

$$\frac{\tan\alpha}{\sec\alpha}=\sin\alpha$$ The trigonometric equation is proved to be an identity by simplifying the left side, using the Quotient and Reciprocal Identities.

Work Step by Step

$$\frac{\tan\alpha}{\sec\alpha}=\sin\alpha$$ We would start from the left side, since it is more complicated. $$A=\frac{\tan\alpha}{\sec\alpha}$$ - Quotient Identity: $$\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$$ - Reciprocal Identity: $$\sec\alpha=\frac{1}{\cos\alpha}$$ Apply them into $A$, we have $$A=\frac{\frac{\sin\alpha}{\cos\alpha}}{\frac{1}{\cos\alpha}}$$ $$A=\frac{\sin\alpha}{\cos\alpha}\times\cos\alpha$$ $$A=\sin\alpha$$ The left side is hence proved to be equal with the right side. The trigonometric equation is an identity.
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