QJEEM21Ma3ee6: JEE Main 2021 Mathematics - Class 12 JEEM21Ma3ee6 Solution: If , then at is :

Question

Verified Solution • 2025–26 Session moderate

If

y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right),x \in \left( {{\pi \over 2},\pi } \right)

, then

{{dy} \over {dx}}

at

x = {{5\pi } \over 6}

is :

Accepted Answer

We have,

y(x)=\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)

=\cot ^{-1} \frac{\left|\cos \frac{x}{2}+\sin \frac{x}{2}\right|+\left|\cos \frac{x}{2}-\sin \frac{x}{2}\right|}{\left|\cos \frac{x}{2}+\sin \frac{x}{2}\right|-\left|\cos \frac{x}{2}-\sin \frac{x}{2}\right|}

\left[\text { as } \cos ^2 \frac{x}{2}+\sin ^2 \frac{x}{2}=1\right.

and

\left.\sin x=2 \sin \frac{x}{2} \cos \frac{x}{2}\right]

\begin{aligned}&amp; =\cot ^{-1}\left(\frac{\cos \frac{x}{2}+\sin \frac{x}{2}+\sin \frac{x}{2}-\cos \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}-\sin \frac{x}{2}+\cos \frac{x}{2}}\right) \forall x \in\left(\frac{\pi}{2}, \pi\right) \\&amp;\end{aligned}

=\cot ^{-1}\left(\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}\right)=\cot ^{-1}\left(\tan \frac{x}{2}\right)

=\frac{\pi}{2}-\tan ^{-1}\left(\tan \frac{x}{2}\right)

\begin{aligned}&amp; \therefore y^{\prime}(x)=\frac{\pi}{2}-\frac{x}{2} \\\\&amp; \Rightarrow \frac{d y}{d x}=y^{\prime}(x)=-\frac{1}{2}=y^{\prime}\left(\frac{5 \pi}{6}\right)\end{aligned}

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