In a triangle `ABC` if `b+c=3a` then find the value of `cot(B/2)cot(C/2)`

Explanation

To solve the problem, we need to find the value of \( \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\) given that \( b + c = 3a \) in triangle \( ABC \).

### Step-by-Step Solution:

1. **Understanding the semi-perimeter**:
The semi-perimeter \( S \) of triangle \( ABC \) is given by:
\[
S = \frac{a + b + c}{2}
...