Question

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

Answer 1

L.H.S.

$\frac{\cos (\pi +x)\cos (-x)}{\sin (\pi -x)\cos (\frac{\pi }{2}+x)}$

$\Rightarrow \frac{-\cos x\left(\cos x\right)}{\sin x(-\sin x)}\therefore \cos (\pi +\theta )=-\cos \theta ,\sin (\pi -\theta )=\sin \theta$

$\Rightarrow \frac{{\cos }^{2}x}{{\sin }^{2}x}$

$\Rightarrow {\cot }^{2}x$

R.H.S.

Hence, proved.