Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If $3a=b+c$

then $\tan \frac{B}{2}\times \tan \frac{C}{2}$ is equal to?

• A. $\tan \frac{A}{2}$
• B. $1$
• C. $2$
• D. none of these

Answer 1

The correct option is D none of these

$tan\frac{B}{2}\times tan\frac{C}{2}=\frac{{△}^2}{s^2(s-b)(s-c)}=\frac{{△}^2(s-a)}{s^2(s-a)(s-b)(s-c)}=\frac{s-a}{s}$

we know, $s=\frac{a+b+c}{2}$

now, it is given that $3a=b+c$

so, $s=\frac{a+3a}{2}=2a$

So, $\frac{s-a}{s}=\frac{2a-a}{2a}=\frac{1}{2}$

Therefore, Answer is $D$