If `y=cot^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))]`, `(0 lt x lt pi/2)`, then `(dy)/(dx)=`

Explanation

We have,
$$
\begin{aligned}
&\Rightarrow \cot ^{-1}\left(\frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}}\right) \\
&\Rightarrow \cot ^{-1}\left(\frac{(\sqrt{1-\sin x}+\sqrt{1+\sin x})^{2}}{(1-\sin x)-(1+\sin x)}\right) \\
&\Rightarrow \cot ^{-1}\left(\frac{1-\sin x+1+\sin x+2 \sqrt{1-\sin ^{2} x}}{1-\sin x-1-\sin x}\right) \\
&\Rightarrow \cot ^{-1}\left(\frac{2+2 \sqrt{\cos ^{2} x}}{-2 \sin x}\right) \\
&\Rightarrow \cot ^{-1}\left(\frac{1+\cos x}{-\sin x}\right) \\
...