In a $ riangle A B C$, if $3 a=b+c$, then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

Question

In a $	riangle A B C$, if $3 a=b+c$, then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

Aicharya · Accepted Answer

We have $\cot \frac{C}{2} \cot \frac{B}{2}$ $=\sqrt{\frac{s(s-c)}{(s-a)(s-b)}} \cdot \sqrt{\frac{s(s-b)}{(s-a)(s-c)}}=\frac{s}{s-a}$ $=\frac{2 s}{2 s-2 a}=\frac{a+b+c}{a+b+c-2 a}$ $=\frac{a+3 a}{3 a-a}=\frac{4 a}{2 a}=2 \quad[\because b+c=3 a]$

In a $\triangle A B C$, if $3 a=b+c$, then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

Solution