Let $f \theta=\sin \left\{\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right\}$, where $-\frac{\pi}{4}<\theta<\frac{\pi}{4}$. Then the value of $\frac{d}{d(\tan \theta)}(f(\theta))$, is

1

We have,

$f(\theta)=\sin \left\{\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{1-2 \sin ^2 \theta}}\right)\right\}$

$\Rightarrow f(\theta)=\sin \left\{\sin ^{-1}\left(\frac{\sin \theta}{\sqrt{\sin ^2 \theta+1-2 \sin ^2 \theta}}\right)\right\}$

$\Rightarrow f(\theta)=\sin \left\{\sin ^{-1}(\tan \theta)\right\}=\tan \theta$

∴  $\frac{d}{d(\tan \theta)}(f(\theta))=1$