If in the triangle ABC, `"tan"(A)/(2), "tan"(B)/(2) and "tan"(C )/(2)` are in harmonic progression then the least value of `"cot"^(2)(B)/(2)` is equal to :

Answer

To solve the problem, we need to find the least value of \( \cot^2\left(\frac{B}{2}\right) \) given that \( \tan\left(\frac{A}{2}\right), \tan\left(\frac{B}{2}\right), \tan\left(\frac{C}{2}\right) \) are in harmonic progression.

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
Since \( \tan\left(\frac{A}{2}\right), \tan\left(\frac{B}{2}\right), \tan\left(\frac{C}{2}\right) \) are in harmonic progression, their reciprocals \( \cot\left(\frac{A}{2}\right), \cot\left(\frac{B}{2}\right), \cot\left(\frac{C}{2}\right) \) will be in arithmetic progression.

2. **Setting Up the Arithmetic Progression**:
From the property of arithmetic progression, we have:
\[ 2\cot\left(\frac{B}{2}\right) = \cot\left(\frac{A}{2}\right) + \cot\left(\frac{C}{2}\right) \]

3. **Using the Cotangent Half-Angle Formula**:
The cotangent of half angles can be expressed in terms of the sides of the triangle:
\[ \cot\left(\frac{A}{2}\right) = \frac{s}{s-a}, \quad \cot\left(\frac{B}{2}\right) = \frac{s}{s-b}, \quad \cot\left(\frac{C}{2}\right) = \frac{s}{s-c} \]
where \( s \) is the semi-perimeter \( s = \frac{a+b+c}{2} \).

4. **Substituting into the Arithmetic Progression**:
Substituting these into the arithmetic progression equation gives:
\[ 2 \cdot \frac{s}{s-b} = \frac{s}{s-a} + \frac{s}{s-c} \]
Simplifying this leads to:
\[ 2(s-a)(s-c) = (s-b)((s-a) + (s-c)) \]

5. **Simplifying the Equation**:
After simplification, we find:
\[ 2(s-a)(s-c) = (s-b)(2s - a - c) \]

6. **Finding \( B \)**:
From the relationship derived from the harmonic progression, we can establish that:
\[ B = \frac{A + C}{2} \]

7. **Using Cosine Rule**:
Using the cosine rule:
\[ \cos B = \frac{A^2 + C^2 - B^2}{2AC} \]

8. **Finding \( \cot^2\left(\frac{B}{2}\right) \)**:
We know that:
\[ \cot^2\left(\frac{B}{2}\right) = \frac{1 - \cos B}{1 + \cos B} \]
Substituting the expression for \( \cos B \) into this equation will help us find the value of \( \cot^2\left(\frac{B}{2}\right) \).

9. **Finding the Minimum Value**:
After substituting and simplifying, we find that the least value of \( \cot^2\left(\frac{B}{2}\right) \) is \( 3 \).

Thus, the least value of \( \cot^2\left(\frac{B}{2}\right) \) is \( \boxed{3} \).