Evaluate the definite integrals`int_0^pi(sin^2(x/2)-cos^2(x/2))dx`

Explanation

To evaluate the definite integral \( \int_0^\pi \left( \sin^2\left(\frac{x}{2}\right) - \cos^2\left(\frac{x}{2}\right) \right) dx \), we can follow these steps:

### Step 1: Substitute \( x = 2t \)
We start by making the substitution \( x = 2t \). This implies that \( dx = 2dt \). The limits of integration change as follows:
- When \( x = 0 \), \( t = 0 \).
- When \( x = \pi \), \( t = \frac{\pi}{2} \).

Thus, the integral becomes:
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