Precalculus (6th Edition)

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

Chapter 7 - Trigonometric Identities and Equations - 7.4 Double-Angle and Half-Angle Identities - 7.4 Exercises: 34

Answer

$\cos x(1-4\sin^{2}x)$ (equivalent answers may differ if we choose other double angle identities for cosine)

Work Step by Step

$\cos 3x=\cos(x+2x)=$ ... Sum identity: $... \cos(A+B)=\cos A\cos B-\sin A\sin B$ $= \cos x\cos 2x-\sin x\sin 2x$ $...$Double angle identities: $...\sin 2A=2\sin A\cos A$ $...\cos 2A=1-2\sin^{2}A$ $=\cos x(1-2\sin^{2}x)-\sin x(2\sin x\cos x)$ $=\cos x-2\sin^{2}x\cos x-2\sin^{2}x\cos x$ $=\cos x-4\sin^{2}x\cos x$ $=\cos x(1-4\sin^{2}x)$ (answers may differ if we choose other double angle identities for cosine, $\cos 2A=\cos^{2}A-\sin^{2}A$, $\cos 2A=2\cos^{2}A- \mathrm{l}$)
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