Let $f( heta)=\sin \left[ an ^{-1}\left(\frac{\sin heta}{\sqrt{\cos 2 heta}} ight) ight]$, where $-\frac{\pi}{4}

Question

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

Aicharya · Accepted Answer

$f(	heta)=\sin \left(	an ^{-1} \frac{\sin 	heta}{\sqrt{\cos 2 	heta}}ight),-\frac{\pi}{4} $ \begin{aligned} 	herefore \quad \sin \phi & =\frac{\sin 	heta}{\sqrt{\sin ^2 	heta+\cos 2 	heta}} \ & =\frac{\sin 	heta}{\sqrt{1-\sin ^2 	heta}}=\frac{\sin 	heta}{\cos 	heta}=	an 	heta \end{aligned} $ $ \begin{array}{ll} 	herefore & f(	heta)=\sin \phi=	an 	heta \ \Rightarrow & \frac{d f(	heta)}{d(	an 	heta)}=1 \end{array} $

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

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