In a `DeltaABC,a-2b+c=0`, then the value of `Cot((A)/(2))Cot((C)/(2))`

Options

A. A. `9//2`
B. B. 3
C. C. `3//2`
D. D. 1

Answer

To solve the problem, we need to find the value of \( \cot\left(\frac{A}{2}\right) \cot\left(\frac{C}{2}\right) \) given that \( a - 2b + c = 0 \).

### Step-by-Step Solution:

1. **Rearranging the Given Equation**:
From the equation \( a - 2b + c = 0 \), we can rearrange it to find a relationship between the sides:
\[ a + c = 2b \]

2. **Using the Semi-Perimeter**:
The semi-perimeter \( S \) of triangle \( ABC \) is given by:
\[ S = \frac{a + b + c}{2} \]
Substituting \( a + c = 2b \) into the semi-perimeter formula:
\[ S = \frac{2b + b}{2} = \frac{3b}{2} \]

3. **Using the Cotangent Half-Angle Formula**:
The cotangent half-angle formulas for a triangle are:
\[ \cot\left(\frac{A}{2}\right) = \sqrt{\frac{S(S - a)}{(S - b)(S - c)}} \]
\[ \cot\left(\frac{C}{2}\right) = \sqrt{\frac{S(S - c)}{(S - b)(S - a)}} \]

4. **Calculating \( S - a \), \( S - b \), and \( S - c \)**:
From \( S = \frac{3b}{2} \):
- \( S - a = \frac{3b}{2} - a \)
- \( S - b = \frac{3b}{2} - b = \frac{b}{2} \)
- \( S - c = \frac{3b}{2} - c \)

5. **Substituting into the Cotangent Formulas**:
Now we can substitute these values into the cotangent formulas:
\[ \cot\left(\frac{A}{2}\right) = \sqrt{\frac{\frac{3b}{2} \left(\frac{3b}{2} - a\right)}{\left(\frac{b}{2}\right) \left(\frac{3b}{2} - c\right)}} \]
\[ \cot\left(\frac{C}{2}\right) = \sqrt{\frac{\frac{3b}{2} \left(\frac{3b}{2} - c\right)}{\left(\frac{b}{2}\right) \left(\frac{3b}{2} - a\right)}} \]

6. **Multiplying the Cotangent Values**:
Now, we multiply \( \cot\left(\frac{A}{2}\right) \) and \( \cot\left(\frac{C}{2}\right) \):
\[ \cot\left(\frac{A}{2}\right) \cot\left(\frac{C}{2}\right) = \sqrt{\frac{\frac{3b}{2} \left(\frac{3b}{2} - a\right)}{\left(\frac{b}{2}\right) \left(\frac{3b}{2} - c\right)}} \cdot \sqrt{\frac{\frac{3b}{2} \left(\frac{3b}{2} - c\right)}{\left(\frac{b}{2}\right) \left(\frac{3b}{2} - a\right)}} \]

7. **Simplifying the Expression**:
After simplification, the terms \( \left(\frac{3b}{2} - a\right) \) and \( \left(\frac{3b}{2} - c\right) \) will cancel out, leading to:
\[ \cot\left(\frac{A}{2}\right) \cot\left(\frac{C}{2}\right) = \frac{3b/2}{b/2} = 3 \]

### Final Answer:
Thus, the value of \( \cot\left(\frac{A}{2}\right) \cot\left(\frac{C}{2}\right) \) is \( 3 \).