$\begin{array}{c}\text { Example 3. Simplify: } \sin \theta \sin \left(\frac{\pi}{2}-\theta\right)+\sin ^{2}\left(\frac{\pi}{4}+\theta\right) \\ -\sin ^{2}\left(\frac{\pi}{6}-\theta\right)-\frac{1}{2} \sin \left(\frac{\pi}{6}+2 \theta\right) .\end{array}$
$\begin{array}{l}\text { Solve the original expression }=\sin \theta \cos \theta+\sin \frac{5 \pi}{12} \sin \left(\frac{\pi}{12}+2 \theta\right) \\ -\frac{1}{2} \sin \left(\frac{\pi}{6}+2 \theta\right) \\ =\frac{1}{2}\left[\sin 2 \theta-\sin \left(\frac{\pi}{6}+2 \theta\right)\right] \\ +\cos \frac{\pi}{12} \sin \left(\frac{\pi}{12}+2 \theta\right) \\ =-\cos \left(\frac{\pi}{12}+2 \theta\right) \sin \frac{\pi}{12} \\ +\cos \frac{\pi}{12} \sin \left(\frac{\pi}{12}+2 \theta\right) \\ =\sin 2 \theta, \\\end{array}$