Question

Prove that $\frac{\cos (\pi +x)\cos (-x)}{\sin (\pi -x)\cos (\frac{\pi }{2}+x)}={\cot }^{2}x$

Answer 1

$L.H.S.=\frac{\cos (\pi +x)\cos (-x)}{\sin (\pi -x)\cos (\frac{\pi }{2}+x)}$
$=\frac{(-\cos x)\cos x}{\sin x(-\sin x)}$
=$\frac{{\cos }^{2}x}{{\sin }^{2}x}$
=${\cot }^{2}x=R.H.S.$
Hence proved.