Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 5 - Section 5.2 - Sum and Difference Formulas - Exercise Set - Page 669: 11



Work Step by Step

Evaluate the term $\cos \left( \alpha -\beta \right)$ using the cosines difference formula and solve the expression on the left-hand side of the identity as, $\cos \left( x-\frac{\pi }{4} \right)=\cos x\cos \frac{\pi }{4}+\sin x\sin \frac{\pi }{4}$ Substitute the values $\cos \frac{\pi }{4}=\frac{\sqrt{2}}{2}\text{ and }\sin \frac{\pi }{4}=\frac{\sqrt{2}}{2}$. $\begin{align} & \cos \left( x-\frac{\pi }{4} \right)=\cos x\left( \frac{\sqrt{2}}{2} \right)+\sin x\left( \frac{\sqrt{2}}{2} \right) \\ & =\frac{\sqrt{2}}{2}\left( \cos x+\sin x \right) \end{align}$ Since the left side part of the identity is equivalent to the expression on the right side, therefore, the identity is verified.
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