Answer
See below.
Work Step by Step
We know that $\cos(2\theta)=\cos^2\theta-\sin^2\theta$ and that $\sin(2\theta)=2\sin\theta\cos\theta$, hence here $\cos 4\theta=\cos^2(2\theta)-\sin^2(2\theta)=(\cos^2\theta-\sin^2\theta)^2-(2\sin\theta\cos\theta)^2=\cos^4\theta-\sin^4\theta-2\sin^2\theta\cos^2\theta-4\sin^2\theta\cos^2\theta=\cos^4\theta-\sin^4\theta-6\sin^2\theta\cos^2\theta$
Thus we proved what we had to.