MCQ
If $\tan (\cot x) = \cot (\tan x),$ then $\sin 2x =$
  • A
    $(2n + 1)\frac{\pi }{4}$
  • $\frac{4}{{(2n + 1)\pi }}$
  • C
    $4\pi (2n + 1)$
  • D
    None of these

Answer

Correct option: B.
$\frac{4}{{(2n + 1)\pi }}$
b
(b)$\tan (\cot x) = \cot (\tan x)$

$ \Rightarrow $ $\tan (\cot x) = \tan \left( {\frac{\pi }{2} - \tan x} \right)$ 

$ \Rightarrow $ $\cot x = n\pi  + \frac{\pi }{2} - \tan x $

$\Rightarrow \cot x + \tan x = n\pi  + \frac{\pi }{2}$

$ \Rightarrow $ $\frac{2}{{\sin 2x}} = n\pi  + \frac{\pi }{2}$

$\Rightarrow \sin 2x = \frac{2}{{n\pi  + \frac{\pi }{2}}} = \frac{4}{{(2n + 1)\pi }}$

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