Question

Let $ A,\,\,B $ and $ C $ are the angles of a plain triangle and $ \tan \left( \frac{A}{2} \right)=\frac{1}{3},\,\,\tan \left( \frac{B}{2} \right)=\frac{2}{3} $ . Then $ \tan \left( \frac{C}{2} \right) $ is equal to:

• A. $ 1/3 $
• B. $ 2/3 $
• C. $ 2/9 $
• D. $ 7/9 $

Answer 1

Answer: (D)

Since, $ A,\,B $ and $ C $ are the angles of a triangle
$ \therefore A+B+C=\pi $
$ \Rightarrow \frac{A}{2}+\frac{B}{2}=\frac{\pi }{2}-\frac{C}{2} $
$ \Rightarrow \tan \left( \frac{A}{2}+\frac{B}{2} \right)=\tan \left( \frac{\pi }{2}-\frac{C}{2} \right) $
$ \Rightarrow \cot \frac{C}{2}=\frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}\tan \frac{B}{2}} $
$ =\frac{\frac{1}{3}+\frac{2}{3}}{1-\frac{1}{3}\times \frac{2}{3}}=\frac{\frac{3}{3}}{\frac{7}{9}}=\frac{9}{7} $
$ \Rightarrow \tan \frac{C}{2}=\frac{7}{9} $