Question

Find the derivative of $y=\ln \tan \left(x/2\right)$

Answer 1

$y=ln\tan \frac{x}{2}$

$\frac{dy}{dx}=\frac{1}{\tan \frac{x}{2}}\cdot {\sec }^2\frac{x}{2}\cdot \frac{1}{2}$ [chain Rule]

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

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

$=\frac{1}{2\sin \frac{x}{2}\cos \frac{x}{2}}[\because \sin 2\theta =2\sin \theta \cos \theta ]$

$=\frac{1}{\sin \left(2\frac{x}{2}\right)}$

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