Verify the identity: $2\cos^4 x - \cos^2 x - 2 \sin^2 x\cos^2 x + \sin^2 x = \cos^2 2x$

Question

Verify the identity: $$2\cos^4 x - \cos^2 x - 2 \sin^2 x\cos^2 x + \sin^2 x = \cos^2 2x$$

There are so many routes to start on this. I have tried a bunch, have gotten stumped at each one.

Answer 1

$$2\cos^4 x - \cos^2 x - 2 \sin^2 x\cos^2 x + \sin^2 x$$

$$=\cos^2 x(2\cos^2 x-1)-\sin^2 x(2\cos^2 x-1)$$

$$=(2\cos^2 x-1)(\cos^2 x-\sin^2 x)$$

Now apply Double-Angle Formulas