Evaluate the definite integral $\int_{0}^{\pi}\left(\sin ^{2} \frac{x}{2}-\cos ^{2} \frac{x}{2}\right) d x$.

(C) Let $I = \int_{0}^{\pi} \left(\sin ^{2} \frac{x}{2} - \cos ^{2} \frac{x}{2}\right) d x$.
We know the trigonometric identity $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$.
Therefore,$\cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}$.
Substituting this into the integral,we get:
$I = \int_{0}^{\pi} -(\cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}) d x$
$I = -\int_{0}^{\pi} \cos x d x$.
The integral of $\cos x$ is $\sin x$.
$I = -[\sin x]_{0}^{\pi}$
$I = -(\sin \pi - \sin 0)$
Since $\sin \pi = 0$ and $\sin 0 = 0$,
$I = -(0 - 0) = 0$.