Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.6 Half-Angle Identities - 5.6 Exercises - Page 243: 50


$$\tan\frac{\theta}{2}=\csc\theta-\cot\theta$$ The equation is an identity, as proved below.

Work Step by Step

$$\tan\frac{\theta}{2}=\csc\theta-\cot\theta$$ We examine the right side first. $$X=\csc\theta-\cot\theta$$ - Reciprocal Identity: $\csc\theta=\frac{1}{\sin\theta}$ - Quotient Identity: $\cot\theta=\frac{\cos\theta}{\sin\theta}$ Apply the identities to $X$: $$X=\frac{1}{\sin\theta}-\frac{\cos\theta}{\sin\theta}$$ $$X=\frac{1-\cos\theta}{\sin\theta}$$ - Half-angle Identity for tangent: $\frac{1-\cos\theta}{\sin\theta}=\tan\frac{\theta}{2}$ Therefore, $$X=\tan\frac{\theta}{2}$$ So 2 sides are equal. $$\tan\frac{\theta}{2}=\csc\theta-\cot\theta$$ The equation is an identity.
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