Q45 / 125 · BITSAT 2016 · Mathematics

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

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

Solution

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

Hence, the correct answer is option: 'B'.

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