Find the range of the function f(x) =x Si x-`(1)/(2) sin^(2)` x for xin `(0,(pi)/(2))`

Explanation

To find the range of the function \( f(x) = x \sin x - \frac{1}{2} \sin^2 x \) for \( x \in (0, \frac{\pi}{2}) \), we will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative \( f'(x) \) using the product rule and the chain rule.

\[ f'(x) = \frac{d}{dx}(x \sin x) - \frac{d}{dx}\left(\frac{1}{2} \sin^2 x\right) \]
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