Prove that

To Prove: $\sin 2 x(\tan x+\cot x)=2$

Taking LHS,

sin 2x(tan x + cot x)

We know that,

$\tan \theta=\frac{\sin \theta}{\cos \theta} \& \cot \theta=\frac{\cos \theta}{\sin \theta}$

$=\sin 2 x\left(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\right)$

$=\sin 2 x\left(\frac{\sin x(\sin x)+\cos x(\cos x)}{\cos x \sin x}\right)$

$=\sin 2 x\left(\frac{\sin ^{2} x+\cos ^{2} x}{\cos x \sin x}\right)$

We know that,

$\sin 2 x=2 \sin x \cos x$

$=2 \sin x \cos x\left(\frac{\sin ^{2} x+\cos ^{2} x}{\cos x \sin x}\right)$

$=2\left(\sin ^{2} x+\cos ^{2} x\right)$

$=2 \times 1\left[\because \cos ^{2} \theta+\sin ^{2} \theta=1\right]$

$=2$

= RHS

∴ LHS = RHS

Hence Proved