Step 1: Understanding the Concept:
Identify the trigonometric identity. We know that $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$. Therefore, $\cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}$.
Step 2: Formula Derivation:
The integrand is the negative of the $\cos x$ identity:
$$\left( \sin^{2} \frac{x}{2} - \cos^{2} \frac{x}{2} \right) = - \left( \cos^{2} \frac{x}{2} - \sin^{2} \frac{x}{2} \right) = -\cos x$$
Step 3: Explanation:
Now, integrate $-\cos x$ from $0$ to $\pi$:
$$\int_{0}^{\pi} -\cos x \, dx = [-\sin x]_{0}^{\pi}$$
$$(-\sin \pi) - (-\sin 0) = 0 - 0 = 0$$
Step 4: Final Answer:
The correct option is (a).