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 - Lesson Check - Page 908: 2



Work Step by Step

Step 1: Replace $\csc\theta$ with $\frac{1}{\sin\theta}$ and $\cot\theta$ with $\frac{\cos\theta}{\sin\theta}$ Hence, $\csc^{2}\theta-\cot^{2}\theta=\frac{1}{\sin^{2}\theta}-\frac{\cos^{2}\theta}{\sin^{2}\theta}$ Step 2: Combine numerators with same denominator, hence, $\csc^{2}\theta-\cot^{2}\theta = \frac{1-\cos^{2}\theta}{\sin^{2}\theta}$ Step 3: Since $\sin^{2}\theta+\cos^{2}\theta=1$, replace $(1-cos^{2}\theta)$ with $sin^{2}\theta$ Hence, $\csc^{2}\theta-\cot^{2}\theta = \frac{\sin^{2}\theta}{\sin^{2}\theta}=1$
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