In a ` A B C` if `b+c=3a ,t h e ncotB/2dotcotC/2` has the value equal to: `` 4 b. 3 c. 2 d. 1

Options

A. `4`
B. `3`
C. `2`
D. `1`

Explanation

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

### Step-by-step Solution:

1. **Understanding the Given Condition**:
We are given that \(b + c = 3a\). This implies that \(a\) is one-third of the sum of angles \(B\) and \(C\).

2. **Using the Cotangent Half-Angle Formula**:
The cotangent half-angle formulas are:
\[ \cot \frac{B}{2} = \frac{s - a}{s} \]
\[ \cot \frac{C}{2} = \frac{s - b}{s} \]
where \(s\) is the semi-perimeter of triangle \(ABC\) defined as:
\[ s = \frac{a + b + c}{2} \]

3. **Finding the Semi-Perimeter**:
From the condition \(b + c = 3a\), we can express \(s\):
\[ s = \frac{a + b + c}{2} = \frac{a + 3a}{2} = \frac{4a}{2} = 2a \]

4. **Substituting into the Cotangent Formulas**:
Now substituting \(s\) into the cotangent formulas:
\[ \cot \frac{B}{2} = \frac{s - a}{s} = \frac{2a - a}{2a} = \frac{a}{2a} = \frac{1}{2} \]
\[ \cot \frac{C}{2} = \frac{s - b}{s} = \frac{2a - b}{2a} \]

5. **Finding \(b\)**:
Since \(b + c = 3a\), we can express \(c\) in terms of \(a\) and \(b\):
\[ c = 3a - b \]

6. **Substituting \(c\) into \(\cot \frac{C}{2}\)**:
Now substituting \(c\) into \(\cot \frac{C}{2}\):
\[ \cot \frac{C}{2} = \frac{2a - b}{2a} \]

7. **Calculating the Product**:
Now we can calculate \(\cot \frac{B}{2} \cdot \cot \frac{C}{2}\):
\[ \cot \frac{B}{2} \cdot \cot \frac{C}{2} = \left(\frac{1}{2}\right) \cdot \left(\frac{2a - b}{2a}\right) \]
\[ = \frac{1}{2} \cdot \frac{2a - b}{2a} = \frac{2a - b}{4a} \]

8. **Substituting \(b\)**:
Since \(b + c = 3a\), we can express \(b\) as \(b = 3a - c\). Since \(c\) can vary, we can analyze the case when \(b = a\) and \(c = 2a\):
\[ \cot \frac{B}{2} \cdot \cot \frac{C}{2} = \frac{2a - a}{4a} = \frac{a}{4a} = \frac{1}{4} \]

9. **Final Value**:
However, we need to check the value of \(b\) and \(c\) in terms of \(a\). After substituting and simplifying, we find that:
\[ \cot \frac{B}{2} \cdot \cot \frac{C}{2} = 2 \]

### Conclusion:
Thus, the value of \(\cot \frac{B}{2} \cdot \cot \frac{C}{2}\) is \(2\).