Answer
See the proof below.
Work Step by Step
Consider the given expression $\frac{\cos 2x}{\cos x-\sin x}$ of the identity and solve it.
Recall the identity $\cos 2x={{\cos }^{2}}x-{{\sin }^{2}}x$ and the expansion of ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$.
$\begin{align}
& \frac{\cos 2x}{\cos x-\sin x}=\frac{{{\cos }^{2}}x-{{\sin }^{2}}x}{\cos x-\sin x} \\
& =\frac{\left( \cos x+\sin x \right)\left( \cos x-\sin x \right)}{\cos x-\sin x} \\
& =\cos x+\sin x
\end{align}$
Hence, proved.
