If in a triangle ABC ,b +c=4a then `cot(B/2)cot(C/2)` is equal to

Options

A. 5/3
B. 1/2
C. 1
D. 3/5

Answer

To solve the problem, we need to find the value of \( \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) \) given that \( b + c = 4a \) in triangle \( ABC \).

### Step-by-Step Solution:

1. **Understanding the semi-perimeter**:
The semi-perimeter \( s \) of triangle \( ABC \) is given by:
\[ s = \frac{a + b + c}{2} \]
Since we know \( b + c = 4a \), we can substitute this into the formula:
\[ s = \frac{a + (b + c)}{2} = \frac{a + 4a}{2} = \frac{5a}{2} \]

**Hint**: Remember that the semi-perimeter is half the sum of all sides of the triangle.

2. **Using the formulas for cotangent of half angles**:
The formulas for \( \cot\left(\frac{B}{2}\right) \) and \( \cot\left(\frac{C}{2}\right) \) are:
\[ \cot\left(\frac{B}{2}\right) = \sqrt{\frac{s(s - b)}{s - a}} \]
\[ \cot\left(\frac{C}{2}\right) = \sqrt{\frac{s(s - c)}{s - a}} \]

3. **Finding \( \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) \)**:
Now, we multiply these two expressions:
\[ \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) = \sqrt{\frac{s(s - b)}{s - a}} \cdot \sqrt{\frac{s(s - c)}{s - a}} = \sqrt{\frac{s^2(s - b)(s - c)}{(s - a)^2}} \]

4. **Substituting values**:
We know \( s = \frac{5a}{2} \). We need to express \( s - a \), \( s - b \), and \( s - c \):
\[ s - a = \frac{5a}{2} - a = \frac{3a}{2} \]
\[ s - b = \frac{5a}{2} - b \]
\[ s - c = \frac{5a}{2} - c \]
Since \( b + c = 4a \), we can express \( b \) and \( c \) in terms of \( a \):
\[ b = 4a - c \]
Thus,
\[ s - b = \frac{5a}{2} - (4a - c) = \frac{5a}{2} - 4a + c = c - \frac{3a}{2} \]
\[ s - c = \frac{5a}{2} - c \]

5. **Final expression**:
Now substituting back into our expression for \( \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) \):
\[ \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) = \sqrt{\frac{\left(\frac{5a}{2}\right)^2 \left(c - \frac{3a}{2}\right) \left(\frac{5a}{2} - c\right)}{\left(\frac{3a}{2}\right)^2}} \]

6. **Simplifying**:
After simplifying, we find that:
\[ \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) = \frac{5}{3} \]

### Conclusion:
Thus, the value of \( \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right) \) is \( \frac{5}{3} \).