Question

Prove the following:
$\cos (\frac{3\pi }{2}+x)\cos (2\pi +x)[\cot (\frac{3\pi }{2}-x)+\cot (2\pi +x\left)\right]=1$

Answer 1

$LHS=cos(\frac{3\pi }{2}+x)cos(2\pi +x)[cot(\frac{3\pi }{2}-x)+cot(2\pi +x\left)\right]$

$=\sin x\cos x[\tan x+\cot x]$

$=\frac{\sin x\cos x({\sin }^{2}x+{\cos }^{2}x)}{\sin x\cos x}=1=RHS$

Hence proved