Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

$f(\theta )=\sin \left({\tan }^{-1}\frac{\sin \theta }{\sqrt{\cos 2\theta }}\right),-\frac{\pi }{4}<\theta <\frac{\pi }{4}$ Let ${\tan }^{-1}\frac{\sin \theta }{\sqrt{\cos 2\theta }}=\varphi$ $$\Rightarrow \tan \varphi =\frac{\sin \theta }{\sqrt{\cos 2\theta }}$$
$$\begin{array}{rl}\therefore \sin \varphi & =\frac{\sin \theta }{\sqrt{{\sin }^2\theta +\cos 2\theta }}\\ & =\frac{\sin \theta }{\sqrt{1-{\sin }^2\theta }}=\frac{\sin \theta }{\cos \theta }=\tan \theta \end{array}$$ $$\begin{array}{ll}\therefore & f(\theta )=\sin \varphi =\tan \theta \\ \Rightarrow & \frac{df(\theta )}{d(\tan \theta )}=1\end{array}$$