Question

\( \int_{0}^{\pi} \left(\sin^2 \frac{x}{2} - \cos^2 \frac{x}{2}\right) dx =\) _____

• A. \( 0 \)
• B. \( -1 \)
• C. \( 1 \)
• D. \( 2 \)

Hint:

Use trigonometric identities to simplify expressions before integrating, especially in definite integrals.

Answer 1

The Correct Option is A

Concept: Use identity: \[ \sin^2 A - \cos^2 A = -\cos(2A) \]
Step 1: Apply identity. \[ \sin^2 \frac{x}{2} - \cos^2 \frac{x}{2} = -\cos x \]
Step 2: Rewrite the integral. \[ \int_{0}^{\pi} (-\cos x) dx = -\int_{0}^{\pi} \cos x \, dx \]
Step 3: \[ = -[\sin x]_0^{\pi} = -(0 - 0) = 0 \]