Mathematics - Differentiation Question with Solution | TestHub

We have $\frac{d}{dx}\left({\tan }^{-1}\sqrt{\frac{1+\cos \frac{x}{2}}{1-\cos \frac{x}{2}}}\right)$ $=\frac{d}{dx}\left[{\tan }^{-1}\sqrt{\frac{1+\left(2{\cos }^2\frac{x}{4}-1\right)}{1-\left(1-2{\sin }^2\frac{x}{4}\right)}}\right]$ $=\frac{d}{dx}({\tan }^{-1}\sqrt{\frac{2{\cos }^2\frac{x}{4}}{2{\sin }^2\frac{x}{4}})}$ $=\frac{d}{dx}\left({\tan }^{-1}\sqrt{{\cot }^2\frac{x}{4}}\right)=\frac{d}{dx}\left({\tan }^{-1}\cot \frac{x}{4}\right)$ $=\frac{d}{dx}\left({\tan }^{-1}\tan \left(\frac{\pi }{2}-\frac{x}{4}\right)\right)=\frac{d}{dx}\left(\frac{\pi }{2}-\frac{x}{4}\right)=0-\frac{1}{4}=-\frac{1}{4}$