Question

Differentiate the following: ${\cot }^{-1}\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{(1+\sin x)}-\sqrt{(1-\sin x)}}$

• A. $\frac{1}{2}.$
• B. $\frac{-1}{2}.$
• C. $\frac{1}{4}.$
• D. $\frac{-1}{4}.$

Answer 1

The correct option is A $\frac{1}{2}.$

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

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

or $y={\cot }^{-1}\cot \frac{x}{2}=\frac{x}{2},$

$\therefore \frac{dy}{dx}=\frac{1}{2}.$
Note: Here assumption taken that $\tan \frac{x}{2}\ge 1$