Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 7 - Trigonometric Identities and Equations - Chapter 7 Test Prep - Review Exercises - Page 737: 51


$\frac{\sin^2 x}{2-2\cos x}=\cos^2\frac{x}{2}$

Work Step by Step

Simplify the left side: $\frac{\sin^2 x}{2-2\cos x}$ $=\frac{1-\cos^2 x}{2(1-\cos x)}$ $=\frac{(1+\cos x)(1-\cos x)}{2(1-\cos x)}$ $=\frac{1+\cos x}{2}$ Simplify the right side: $=\cos^2 \frac{x}{2}$ $=(\pm\sqrt{\frac{1+\cos x}{2}})^2$ $=\frac{1+\cos x}{2}$ Since the left side and the right side simplify to the same expression, the expressions are equal and the identity is proven.
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