Answer:
Option B
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 , $\sin x+\cos x= \sin \frac{\pi}{4}+\cos \frac {\pi}{4}$
$=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2}$
