If y(x)=^-1(√1+ x+√1- x√1+ x-√1- x), x(2,), then dydx at x=56 is

Question

If $y(x)=\cot^{-1}\left(\dfrac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$, $x\in\left(\dfrac{\pi}{2},\pi\right)$, then $\dfrac{dy}{dx}$ at $x=\dfrac{5\pi}{6}$ 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 $\left.\sin x=2 \sin \frac{x}{2} \cos \frac{x}{2}\right]$

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

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

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

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

Solutions · 1