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 - Page 909: 42


$$\sqrt {\frac{1}{\cos^{2}\theta}-1}$$

Work Step by Step

Start with the Pythagorean Identity $$\sin^{2}\theta+\cos^{2}\theta = 1.$$ Rearrange terms and solve for $\sin\theta$: $$\sin^{2}\theta=1-\cos^{2}\theta$$ $$\sin\theta=\sqrt {1-\cos^{2}\theta}.$$ Next use the Tangent Identity and substitute in the above expression for $\sin\theta$: $$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{\sqrt {1-\cos^{2}\theta}}{\cos\theta}=\sqrt {\frac{1}{\cos^{2}\theta}-1}.$$
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