Question

Differentiate $\log \sqrt{\frac{1+\cos x}{1-\cos x}}w.r.t.x$

Answer 1

$\frac{d}{dx}\log \sqrt{\frac{1+\cos x}{1-\cos x}}$

$1+\cos x=2{\cos }^2\frac{x}{2}$

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

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

$=\frac{d}{dx}\log \tan \frac{x}{2}$

$=\frac{1}{\tan \frac{x}{2}}{\sec }^2\frac{x}{2}\frac{1}{2}$

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

$=\frac{1}{\sin x}$

$=\csc x$