In ` A B C ,(cot A/2+cotB/2)(asin^2B/2+bsin^2A/2)=` `cotC` (b) `ccotC` (c) `cotC/2` (d) `ccotC/2`

Explanation

To solve the problem, we need to evaluate the expression \((\cot \frac{A}{2} + \cot \frac{B}{2})(a \sin^2 \frac{B}{2} + b \sin^2 \frac{A}{2})\) and show that it equals \(c \cot C\).

### Step-by-Step Solution

1. **Express Cotangent in Terms of Sides and Semiperimeter**:
We start by expressing \(\cot \frac{A}{2}\) and \(\cot \frac{B}{2}\) using the formula:
\[
\cot \frac{A}{2} = \frac{\sqrt{s(s-a)}}{\sqrt{s(s-b)}\sqrt{s(s-c)}}
...