Let $f( heta)=\sin \left( an ^{-1}\left(\frac{\sin heta}{\sqrt{\cos 2 heta}} ight) ight)$, where $\frac{-\pi}{4} \lt heta \lt \frac{\pi}{4}$, then the value of $\frac{d}{d( an heta)}(f( heta))$ is

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Let $f(	heta)=\sin \left(	an ^{-1}\left(\frac{\sin 	heta}{\sqrt{\cos 2 	heta}}ight)ight)$, where $\frac{-\pi}{4} \lt 	heta \lt \frac{\pi}{4}$, then the value of $\frac{d}{d(	an 	heta)}(f(	heta))$ is

Aicharya · Accepted Answer

$\begin{aligned} & f(	heta)=\sin \left(	an ^{-1}\left(\frac{\sin 	heta}{\sqrt{\cos 2 	heta}}ight)ight) \ &=\sin \left(\sin ^{-1}\left(\frac{\sin 	heta}{\sqrt{\cos 2 	heta+\sin ^2 	heta}}ight)ight) \ &=\sin \left(\sin ^{-1}\left(\frac{\left.\cdots 	an ^{-1} x=\sin ^{-1} \frac{x}{\sqrt{1+x^2}}ight]}{\sqrt{\cos ^2 	heta-\sin ^2 	heta+\sin ^2 	heta}}ight)ight) \ &=\sin \left(\sin ^{-1}\left(\frac{\sin 	heta}{\sqrt{\cos ^2 	heta}}ight)ight) \ &=\sin \left(\sin ^{-1}(	an 	heta)ight) \ &=	an 	heta \ & 	herefore \quad \frac{d}{d(	an 	heta)}(f(	heta))=1\end{aligned}$

Let $f(\theta)=\sin \left(\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right)$, where…

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