Question

If $ \tan (\cot x)=\cot (\tan x) $ , then $ \sin 2x $ is equal to

• A. $ \frac{2}{(2n+1)\pi } $
• B. $ \frac{4}{(2n+1)\pi } $
• C. $ \frac{2}{n(n+1)\pi } $
• D. $ \frac{4}{n(n+1)\pi } $

Answer 1

Answer: (B)

Given, $ \tan (\cot x)=\cot (\tan x) $
$ =\tan \left( \frac{\pi }{2}-\tan x \right) $
$ \Rightarrow $ $ \cot x=n\pi +\left( \frac{\pi }{2}-\tan x \right) $
$ \Rightarrow $ $ \cot x+\tan x=n\pi +\frac{\pi }{2} $
$ \therefore $ $ \frac{1}{\sin x\cos x}=\frac{\pi }{2}(2n+1) $
$ \Rightarrow $ $ \sin 2x=\frac{4}{(2n+1)\pi } $