Question

Differentiate w.r.t $x$, the following function:
$\log \sqrt{\frac{1+\cos x}{1-\cos x}}$

Answer 1

Let $y=\log \sqrt{\frac{1+\cos x}{1-\cos x}}$

On differentiating $y$ with respect to $x$

$\frac{dy}{dx}=\frac{d}{dx}\log \sqrt{\frac{2{\cos }^{2}x/2}{2{\sin }^{2}x/2}}$

$=\frac{d}{dx}\log \cot x/2$

$=\frac{1}{\cot x/2}\frac{d}{dx}\cot x/2$

$=\frac{\sin x/2}{\cos x/2}(-\frac{1}{2}\times \frac{1}{{\sin }^{2}x/2})$

$=\frac{-1}{2\sin x/2\cos x/2}=\frac{-1}{\sin 2x/2}$

$=\frac{-1}{\sin x}=\text{-cosec}x$