(a)[3]
Show that, after simplification, $\cos^4 \theta - \sin^4 \theta = \cos 2\theta$.
(b)[6]
Hence determine the exact value of $\displaystyle \int_{-\frac{1}{8}\pi}^{\frac{1}{8}\pi} \left( \cos^4\theta - \sin^4\theta + 4\sin^2\theta\cos^2\theta \right) \, d\theta$.