Question

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____

Answer 1

$\sin \left(\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right)$, where $\theta \in\left(-\frac{\pi}{4}, \frac{\pi}{4}\right)$
$\sin \left(\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{2 \cos ^{2} \theta-1}}\right)\right)$
$=\sin \left(\sin ^{-1}(\tan \theta)\right)=\tan \theta$
$\frac{ d (\tan \theta)}{ d (\tan \theta)}=1$