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(A/2)*cot(C/2)`

Explanation

To solve the problem, we need to find the minimum value of \( \cot\left(\frac{A}{2}\right) \cdot \cot\left(\frac{C}{2}\right) \) 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 Condition**:
Since \( \tan\left(\frac{A}{2}\right), \tan\left(\frac{B}{2}\right), \tan\left(\frac{C}{2}\right) \) are in H.P., we can use the property that if three numbers \( x, y, z \) are in H.P., then \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in A.P.

Therefore, we have:
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