## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 5 - Section 5.2 - Sum and Difference Formulas - Exercise Set - Page 669: 12

#### Answer

See the full explanation below.

#### 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, $\cos \left( x-\frac{5\pi }{4} \right)=\cos x\cos \frac{5\pi }{4}+\sin x\sin \frac{5\pi }{4}$ Substitute the values $\cos \frac{5\pi }{4}=-\frac{\sqrt{2}}{2}\text{ and }\sin \frac{\pi }{4}=-\frac{\sqrt{2}}{2}$. \begin{align} & \cos \left( x-\frac{5\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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