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: 36

Answer

$8\cos^{4}x-8\cos^{2}x+1$ (equivalent answers may differ if we choose other double angle identities for cosine)

Work Step by Step

$\cos 2A=\cos^{2}A-\sin^{2} A$ $\cos 2A=1-2\sin^{2}A$ $\cos 2A=2\cos^{2}A- 1$ -------------- (we'll use the third identity, twice) $\cos 4x=\cos[2(2x)]$ $=2\cos^{2}(2x)- 1$ $=2(\cos 2x)^{2}- 1$ $=2(2\cos^{2}x- 1)^{2}- 1$ .... $(A-B)^{2}=A^{2}-2AB+B^{2}...$ $=2(4\cos^{4}x-4\cos^{2}x+1)-1$ $=8\cos^{4}x-8\cos^{2}x+1$ (equivalent answers may differ if we choose other double angle identities for cosine)
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