In any `Delta ABC` , `cot(A/2),cot(B/2),cot(C/2)` are in `A.P.`, then

Options

A. A.P
B. G.P
C. H.P
D. None

Answer

To solve the problem, we need to show that if \( \cot\left(\frac{A}{2}\right), \cot\left(\frac{B}{2}\right), \cot\left(\frac{C}{2}\right) \) are in arithmetic progression (A.P.), then we can derive a relationship involving the angles of triangle \( ABC \).

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
Since \( \cot\left(\frac{A}{2}\right), \cot\left(\frac{B}{2}\right), \cot\left(\frac{C}{2}\right) \) are in A.P., we can write:
\[ 2 \cot\left(\frac{B}{2}\right) = \cot\left(\frac{A}{2}\right) + \cot\left(\frac{C}{2}\right) \]

2. **Using the Angle Sum Property**:
We know that in triangle \( ABC \), the sum of angles is:
\[ A + B + C = 180^\circ \]
Thus, we can express \( B \) as:
\[ B = 180^\circ - A - C \]

3. **Substituting for \( B \)**:
We can substitute this expression into our A.P. condition:
\[ 2 \cot\left(\frac{180^\circ - A - C}{2}\right) = \cot\left(\frac{A}{2}\right) + \cot\left(\frac{C}{2}\right) \]

4. **Simplifying the Cotangent**:
Using the property \( \cot(90^\circ - x) = \tan(x) \), we can rewrite:
\[ \cot\left(\frac{180^\circ - A - C}{2}\right) = \tan\left(\frac{A + C}{2}\right) \]
Therefore, we have:
\[ 2 \tan\left(\frac{A + C}{2}\right) = \cot\left(\frac{A}{2}\right) + \cot\left(\frac{C}{2}\right) \]

5. **Using Cotangent Addition Formula**:
We know that:
\[ \cot\left(\frac{A}{2}\right) + \cot\left(\frac{C}{2}\right) = \frac{\sin\left(\frac{A}{2}\right) + \sin\left(\frac{C}{2}\right)}{\sin\left(\frac{A}{2}\right) \sin\left(\frac{C}{2}\right)} \]
Thus, we can express our equation as:
\[ 2 \tan\left(\frac{A + C}{2}\right) = \frac{\sin\left(\frac{A}{2}\right) + \sin\left(\frac{C}{2}\right)}{\sin\left(\frac{A}{2}\right) \sin\left(\frac{C}{2}\right)} \]

6. **Final Relationship**:
After simplification, we arrive at:
\[ 3 \tan\left(\frac{A}{2}\right) \tan\left(\frac{C}{2}\right) = 1 \]
This implies:
\[ \tan\left(\frac{A}{2}\right) \tan\left(\frac{C}{2}\right) = \frac{1}{3} \]

### Conclusion:
Thus, we conclude that if \( \cot\left(\frac{A}{2}\right), \cot\left(\frac{B}{2}\right), \cot\left(\frac{C}{2}\right) \) are in A.P., then:
\[ \tan\left(\frac{A}{2}\right) \tan\left(\frac{C}{2}\right) = \frac{1}{3} \]