Question:
Prove that
$\frac{\cos 2 x}{\cos x-\sin x}=\cos x+\sin x$
Solution:
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
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