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 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 219: 72

Answer

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Work Step by Step

$$\cos2x=2\cos^2x-1$$ $\text{Solution:}$ \begin{align*} \text{We know that}\\ \cos2x&=\cos(x+x)\\ &=\cos x\cos x-\sin x\sin x ~~~\text{Cosine Sum Identity}\\ &=\cos x\cos x-\sin^2 x ~~~\because \sin x\sin x=\sin^2 x\\ &=\cos^2 x-\sin^2 x ~~~~~~\because \cos x \cos x=\cos^2x\\ &=\cos^2 x-(1-\cos^2 x) ~~~ \because \sin^2 x=1-\cos^2 x\\ &=\cos^2 x-1+\cos^2 x ~~~~\text{Simplify}\\ &=2\cos^2 x-1 ~~~~~~~~~~~~~~\text{Simplify} \end{align*} Since $\cos2x=2\cos^2x-1$, therefore given equation is an identity.
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