Question

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

• A. $\pi$
• B. $2\pi$
• C. $4\pi(2n + 1)$
• D. None of these

Answer 1

Answer: (B)

$2\pi$

Answer 2

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

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

$\tan x + \cot x = \frac{(2n + 1)\pi}{2}$ ⇒ $\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{(2n + 1)\pi}{2}$

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