Question

\[\int\limits_0^\pi \left( \sin^2 \frac{x}{2} - \cos^2 \frac{x}{2} \right) dx\]

Answer 1

\[Let\ I = \int_0^\pi \left( \sin^2 \frac{x}{2} - \cos^2 \frac{x}{2} \right) d\ x\ . Then, \]
\[I = - \int_0^\pi \cos\ x\ dx\ \left[ \because \cos A = \cos^2 \frac{A}{2} - \sin^2 \frac{A}{2} \right]\]
\[ \Rightarrow I = - \left[ \sin x \right]_0^\pi \]
\[ \Rightarrow I = 0\]