Discussion Forum

 If  $2 \tan ^{-}(\cos x)=\tan^{-1}(2 cosec x)$ then sin x+cosx is equal to

Answer:

Explanation:

 Given ,  $2 \tan ^{-}(\cos x)=\tan^{-1}(2 cosec x)$ 

$\Rightarrow$     $tan^{-1}\frac{2\cos x}{1- \cos^{2} x}=\tan^{-1}\left(\frac{2}{\sin x}\right)$

 $\Rightarrow$   $\frac{2\cos x}{1- \cos^{2} x}=\frac{2}{\ sin x}\Rightarrow \frac {\cos x}{sin ^{2} x}\Rightarrow \frac {1}{\sin x}$

   $\Rightarrow  \frac {\cos x}{sin x}=1$    $[\because \sin x\neq0]$

  $\Rightarrow$      $\tan x=1\Rightarrow x= \frac{\pi}{4}$

Now , $\sin x+\cos x= \sin \frac{\pi}{4}+\cos \frac {\pi}{4}$

  $=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2}$