The derivative of ${\tan }^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)$ with respect to $x$ is

Solution

1. Let $y={\tan }^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)$.
2. Using the identities $1-\cos x=2{\sin }^2(x/2)$ and $1+\cos x=2{\cos }^2(x/2)$, the expression simplifies to ${\tan }^{-1}(\sqrt{{\tan }^2(x/2)})={\tan }^{-1}(\tan (x/2))$.
3. This simplifies the function to $y=\frac{x}{2}$.
4. Differentiating with respect to $x$ gives $\frac{dy}{dx}=\frac{1}{2}$.
5. Hence the answer is (B).