Answer
\[ = \frac{\pi }{4}\]
Work Step by Step
\[\begin{gathered}
\int_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {{{\sin }^2}2\theta d\theta } \hfill \\
\hfill \\
use\,\,the\,\,idenity\,\,{\sin ^2}x\, = \frac{{1 - \cos 2x}}{2} \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
\int_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{1 - \cos 4\theta }}{2}} \,d\theta \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= \,\,\left[ {\frac{\theta }{2} - \frac{{\sin 4\theta }}{8}} \right]_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} \hfill \\
\hfill \\
Fundamental\,\,theorem \hfill \\
\hfill \\
= \,\left( {\frac{\pi }{8} - \frac{{\sin \,\left( \pi \right)}}{8}} \right) - \,\left( {\frac{\pi }{8} - \frac{{\sin \,\left( { - \pi } \right)}}{8}} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \frac{\pi }{8} + \frac{\pi }{8} \hfill \\
\hfill \\
solution \hfill \\
\hfill \\
= \frac{\pi }{4} \hfill \\
\end{gathered} \]
