In a `Delta ABC tan ""A/2, tan ""B/2, tan ""C/2 ` are in H.P., then the vlaue iof `cot ""A/2 cot ""C/2` is :

Options

A. 3
B. 2
C. 1
D. `sqrt3`

Answer

To solve the problem, we need to find the value of \( \cot \frac{A}{2} \cot \frac{C}{2} \) given that \( \tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2} \) are in Harmonic Progression (H.P.).

### Step-by-Step Solution:

1. **Understanding H.P. and A.P. Relationship**:
Since \( \tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2} \) are in H.P., it implies that their reciprocals \( \cot \frac{A}{2}, \cot \frac{B}{2}, \cot \frac{C}{2} \) are in Arithmetic Progression (A.P.).

**Hint**: Remember that if three numbers are in H.P., their reciprocals are in A.P.

2. **Using the A.P. Property**:
If \( \cot \frac{A}{2}, \cot \frac{B}{2}, \cot \frac{C}{2} \) are in A.P., then we can write:
\[ 2 \cot \frac{B}{2} = \cot \frac{A}{2} + \cot \frac{C}{2} \]

**Hint**: The property of A.P. states that the middle term is the average of the other two terms.

3. **Using the Sum of Angles in a Triangle**:
In triangle \( ABC \), we know that:
\[ A + B + C = \pi \]
Therefore, we can use the identity:
\[ \cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2} = \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2} \]

**Hint**: This is a standard result in trigonometry for angles of a triangle.

4. **Rearranging the Equation**:
We need to express \( \cot \frac{A}{2} \cot \frac{C}{2} \). From the A.P. property, we can substitute \( \cot \frac{B}{2} \) into the equation:
\[ \cot \frac{A}{2} + \cot \frac{C}{2} = 2 \cot \frac{B}{2} \]
Now substituting this into the sum of angles equation:
\[ 2 \cot \frac{B}{2} + \cot \frac{B}{2} = \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2} \]
This simplifies to:
\[ 3 \cot \frac{B}{2} = \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2} \]

**Hint**: Combine terms carefully to isolate the product of the cotangents.

5. **Dividing by \( \cot \frac{B}{2} \)**:
Assuming \( \cot \frac{B}{2} \neq 0 \), we can divide both sides by \( \cot \frac{B}{2} \):
\[ \cot \frac{A}{2} \cot \frac{C}{2} = 3 \]

**Hint**: Make sure to check for cases where \( \cot \frac{B}{2} \) might be zero, but in a triangle, this is generally valid.

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