Question

If $A+B+C={180}^{\circ }$, then $\sum \tan \frac{A}{2}\tan \frac{B}{2}$ is equal to:

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is B 1

$\frac{A}{2}+\frac{B}{2}={90}^o-\frac{C}{2}$

$\Rightarrow \frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}\tan \frac{B}{2}}=\cot \frac{C}{2}$, Take tangent both sides

$\Rightarrow \frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}\tan \frac{B}{2}}=\frac{1}{\tan \frac{C}{2}}$

$\Rightarrow \sum \tan \frac{A}{2}\tan \frac{B}{2}=1$