Let $a, b, c$ are the sides of the $\triangle A B C$ and $s$ is the semi-perimeter
$$
s=\frac{a+b+c}{2}
$$

Put the value of Eq. (i), we get
$$
s=\frac{a+3 a}{2}=\frac{4 a}{2}=2 a
$$

Now, $\cot \frac{B}{2} \cot \frac{C}{2}$
$$
\begin{gathered}
=\sqrt{\frac{s(s-b)}{(s-a)(s-c)}} \cdot \sqrt{\frac{s(s-c)}{(s-a)(s-b)}} \\
\left\{\because \cot \frac{A}{2}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}\right\}
\end{gathered}
$$
Put the value of $s$, we get
$$
\begin{aligned}
& =\sqrt{\frac{2 a(2 a-b)}{(2 a-a)(2 a-c)}} \cdot \sqrt{\frac{2 a(2 a-c)}{(2 a-a)(2 a-b)}} \\
& =\sqrt{\frac{2 a(2 a-b)}{a(2 a-c)} \times \frac{2 a(2 a-c)}{a(2 a-b)}}=\sqrt{4}=2
\end{aligned}
$$