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

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

$\therefore f(\theta)=\tan \theta$

$\therefore \frac{d f}{d \tan \theta}=1$