If `cot(theta-alpha),3cottheta,"cot"(theta+alpha)` are in A.P. and `theta` is not an integral multiple of `pi/2,` then the value of `(4sin^2theta)/(3sin^2alpha)=` _________

Explanation

To solve the problem, we need to find the value of \(\frac{4 \sin^2 \theta}{3 \sin^2 \alpha}\) given that \(\cot(\theta - \alpha)\), \(3 \cot \theta\), and \(\cot(\theta + \alpha)\) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
Since \(\cot(\theta - \alpha)\), \(3 \cot \theta\), and \(\cot(\theta + \alpha)\) are in A.P., we can use the property of A.P. which states that:
\[
2 \cdot 3 \cot \theta = \cot(\theta - \alpha) + \cot(\theta + \alpha)
...