if ABC is a triangle and `tan(A/2), tan(B/2), tan(C/2)` are in H.P. Then find the minimum value of `cot(B/2)`

Answer

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

### Step-by-Step Solution:

1. **Understanding the relationship between H.P. and A.P.:**
Since \( \tan\left(\frac{A}{2}\right), \tan\left(\frac{B}{2}\right), \tan\left(\frac{C}{2}\right) \) are in H.P., their reciprocals \( \cot\left(\frac{A}{2}\right), \cot\left(\frac{B}{2}\right), \cot\left(\frac{C}{2}\right) \) are in Arithmetic Progression (A.P.).

\[ 2 \cot\left(\frac{B}{2}\right) = \cot\left(\frac{A}{2}\right) + \cot\left(\frac{C}{2}\right) \]

2. **Using the property of angles in a triangle:**
Since \( A + B + C = 180^\circ \), we can express \( C \) in terms of \( A \) and \( B \):
\[ C = 180^\circ - A - B \]

3. **Applying the cotangent addition formula:**
We know that:
\[ \cot\left(\frac{C}{2}\right) = \cot\left(90^\circ - \frac{A + B}{2}\right) = \tan\left(\frac{A + B}{2}\right) \]

4. **Using the cotangent identity:**
From the cotangent addition formula:
\[ \cot\left(\frac{A + B}{2}\right) = \frac{\cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) - 1}{\cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right)} \]

5. **Substituting back into the A.P. condition:**
We substitute \( \cot\left(\frac{C}{2}\right) \) into the A.P. condition:
\[ 2 \cot\left(\frac{B}{2}\right) = \cot\left(\frac{A}{2}\right) + \tan\left(\frac{A + B}{2}\right) \]

6. **Finding the minimum value:**
By applying the inequality of arithmetic and geometric means, we can derive:
\[ \cot\left(\frac{A}{2}\right) + \cot\left(\frac{C}{2}\right) \geq 2\sqrt{\cot\left(\frac{A}{2}\right) \cot\left(\frac{C}{2}\right)} \]
Thus, we can conclude that:
\[ \cot\left(\frac{B}{2}\right) \geq \sqrt{3} \]

7. **Final result:**
Therefore, the minimum value of \( \cot\left(\frac{B}{2}\right) \) is:
\[ \boxed{\sqrt{3}} \]