Question

prove that cosec 2x+cot 2x=cotx

Answer 1

See the pic.
I tried to make you understand.

Thanks.
Tripathy.

Answer 2

cosec 2x + cot 2x =$\cot x$, proved.

Step-by-step explanation:

L.H.S. = cosec 2x + cot 2x

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

[ ∵ cosec 2A $= \dfrac{1}{\sin 2A}$ and $\cot 2A =\dfrac{\cos 2A}{\sin 2A}$]

=$\dfrac{1}{\sin 2x}(1 +\cos 2x)$

$=\dfrac{1}{\sin 2x}(1 +\cos ^{2} x-\sin ^{2} x)$

[∵ $\cos ^{2} A-\sin ^{2} A=\cos 2A$]

$=\dfrac{1}{\sin 2x}(2\cos ^{2} x)$

$=\dfrac{1}{2\sin x\cos x}(2\cos ^{2} x)$

[ ∵$\sin 2A=2\sin A\cos A$]

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

= $\cot x$

= R.H.S., proved.