Trigonometry (10th Edition)

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

Chapter 5 - Trigonometric Identities - Section 5.1 Fundamental Identities - 5.1 Exercises - Page 196: 83

Answer

$$\frac{\csc\theta}{\cot(-\theta)}=-\sec\theta$$

Work Step by Step

$$A=\frac{\csc\theta}{\cot(-\theta)}$$ - Negative-Angle Identity: $$\cot(-\theta)=-\cot\theta$$ Replace into $A$: $$A=\frac{\csc\theta}{-\cot\theta}$$ - Reciprocal Identity: $$\csc\theta=\frac{1}{\sin\theta}$$ - Quotient Identity: $$\cot\theta=\frac{\cos\theta}{\sin\theta}$$ Replace into $A$: $$A=\frac{\frac{1}{\sin\theta}}{-\frac{\cos\theta}{\sin\theta}}$$ $$A=-\frac{1\times\sin\theta}{\sin\theta\cos\theta}$$ $$A=-\frac{1}{\cos\theta}$$ - Reciprocal Identity: $$\sec\theta=\frac{1}{\cos\theta}$$ Therefore, $$A=-\sec\theta$$
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