Question

Differentiate ${\tan }^{-1}{\left(\frac{1-\cos x}{1+\cos x}\right)}^{\frac{1}{2}}$ with respect to $x$.

Answer 1

let $y={\text{tan}}^{-1}(\frac{1-\cos x}{1+\cos x}{)}^{\frac{1}{2}}={\text{tan}}^{-1}(\frac{2{\sin }^2\frac{x}{2}}{2{\cos }^2\frac{x}{2}}{)}^{\frac{1}{2}}=\frac{x}{2}$

differentiating wrt $x$

$\frac{dy}{dx}=\frac{1}{2}$