Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 14 - Trigonometric Identities and Equations - 14-1 Trigonometric Identities - Practice and Problem-Solving Exercises: 36

Answer

$$\sin\theta$$

Work Step by Step

Simplify $$\csc\theta-\cos\theta\cot\theta.$$ Use the Reciprocal Identity $$\csc\theta=\frac{1}{\sin\theta}$$ and the Cotangent Identity $$\cot\theta=\frac{\cos\theta}{\sin\theta}$$ to obtain $$\csc\theta-\cos\theta\cot\theta=\bigg(\frac{1}{\sin\theta}\bigg)-\cos\theta\bigg(\frac{\cos\theta}{\sin\theta}\bigg)$$ $$=\frac{1-\cos^{2}\theta}{\sin\theta}.$$ Use the Pythagorean Identity $$\sin^{2}\theta+\cos^{2}\theta=1$$ rearranged as $$1-\cos^{2}\theta=\sin^{2}\theta$$ to produce $$\csc\theta-\cos\theta\cot\theta=\frac{\sin^{2}\theta}{\sin\theta}=\sin\theta.$$
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