Prove that

To Prove: $\frac{\cos 2 x}{\cos x-\sin x}=\cos x+\sin x$

Taking LHS,

$=\frac{\cos 2 x}{\cos x-\sin x}$

$=\frac{\cos ^{2} x-\sin ^{2} x}{\cos x-\sin x}\left[\because \cos 2 x=\cos ^{2} x-\sin ^{2} x\right]$

Using, $\left(a^{2}-b^{2}\right)=(a-b)(a+b)$

$=\frac{(\cos x-\sin x)(\cos x+\sin x)}{(\cos x-\sin x)}$

$=\cos x+\sin x$

= RHS

∴ LHS = RHS