Question

Evaluate $\frac{\tan A+\sec A-1}{\tan A-\sec A+1}$ and express in terms of tan$\frac{A}{2}$

Answer 1

$\frac{\tan A+\text{sec}A-1}{\tan A-\text{sec}A+1}=\frac{\sin A+1-\cos A}{\sin A-1+\cos A}$ $=\frac{2\sin \frac{A}{2}\cos \frac{A}{2}+2{\sin }^2\frac{A}{2}}{2\sin \frac{A}{2}\cos \frac{A}{2}-2{\sin }^2\frac{A}{2}}=\frac{1+\tan \frac{A}{2}}{1-\tan \frac{A}{2}}$ $=\tan (\frac{\pi }{4}+\frac{A}{2})$