Answer
.
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.
