Prove that

Solution:

To Prove: cosec 2x + cot 2x = cot x

Taking LHS,

= cosec 2x + cot 2x …(i)

We know that,

$\operatorname{cosec} x=\frac{1}{\sin x} \& \cot x=\frac{\cos x}{\sin x}$

Replacing x by 2x, we get

$\operatorname{cosec} 2 x=\frac{1}{\sin 2 x} \& \cot 2 x=\frac{\cos 2 x}{\sin 2 x}$

So, eq. (i) becomes

$=\frac{1}{\sin 2 x}+\frac{\cos 2 x}{\sin 2 x}$

$=\frac{1+\cos 2 x}{\sin 2 x}$

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

$=\frac{2 \cos ^{2} x}{2 \sin x \cos x}[\because \sin 2 x=2 \sin x \cos x]$

$=\frac{\cos x}{\sin x}$

$=\cot x\left[\because \cot x=\frac{\cos x}{\sin x}\right]$

= RHS

Hence Proved