• # question_answer Let A, B and C are the angles of a plain triangle and $\tan \frac{A}{2}=\frac{1}{3},\,\,\tan \frac{B}{2}=\frac{2}{3}$. Then $\tan \frac{C}{2}$ is equal to   [Orissa JEE 2003] A) 7/9 B) 2/9 C) 1/3 D) 2/3

$A+B+C=\pi$ $\therefore \,\,\,\tan \left( \frac{A+B}{2} \right)=\tan \left( \frac{\pi }{2}-\frac{C}{2} \right)$ Þ $\frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}.\tan \frac{B}{2}}=\cot \frac{C}{2}\Rightarrow \frac{\frac{1}{3}+\frac{2}{3}}{1-\frac{1}{3}.\frac{2}{3}}=\frac{9}{7}=\cot \frac{C}{2}$ \ $\tan \frac{C}{2}=\frac{7}{9}$.