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: 7

Answer

$\cos{\theta}\cot{\theta}=\frac{1}{\sin\theta}-\sin\theta$ Refer to the step-by-step part for the proof.

Work Step by Step

Step 1: Replace $\cot\theta$ by $\frac{\cos\theta}{\sin\theta}$ Hence, $\cos\theta\cot\theta=\cos\theta\times \frac{\cos\theta}{\sin\theta}=\frac{\cos^{2}\theta}{\sin\theta}$ Step 2: By the identity $\sin^{2}\theta+\cos^{2}\theta=1$; replace $\cos^{2}\theta$ by $1-\sin^{2}\theta$. Hence, $\cos\theta\cot\theta=\frac{1-\sin^{2}\theta}{\sin\theta}$ Step 3: Split the denominator across common numerator, $\cos\theta\cot\theta=\frac{1}{\sin\theta}-\frac{\sin^{2}\theta}{\sin\theta}$ Step 4: Cancel one $\sin\theta$ in the second term, Hence, $\cos\theta\cot\theta=\frac{1}{\sin\theta}-\sin\theta$
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