Question

If $A+B+C=\pi$ then,$\tan \frac{A}{2}\cdot \tan \frac{B}{2}+\tan \frac{B}{2}\cdot \tan \frac{C}{2}+\tan \frac{C}{2}\cdot \tan \frac{A}{2}$ =

• A. $\tan \frac{A}{2}$ . $\tan \frac{B}{2}$ . $\tan \frac{C}{2}$
• B. $\tan \frac{A}{2}$ + $\tan \frac{B}{2}$ + $\tan \frac{C}{2}$
• C. $-\tan \frac{A}{2}$. $\tan \frac{B}{2}$ . $\tan \frac{C}{2}$
• D. $1$

Answer 1

The correct option is D

$1$

Given $A+B+C=\pi$

All the terms are half angles

Dividing both sides by 2

$\frac{A+B+C}{2}$$=\frac{\pi }{2}$

$\frac{A}{2}$ + $\frac{B}{2}$ = $\frac{\pi }{2}$ - $\frac{C}{2}$

$\tan (\frac{A}{2}$ + $\frac{B}{2})$ = $\tan (\frac{\pi }{2}$ - $\frac{C}{2})$

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

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

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