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