Q:

Prove that $4\cos^4x-2\cos2x-1/2\cos4x$ is independent of x

$4\cos^4x-2\cos2x-1/2\cos4x$
I don't know how to proceed.  $\cos4x=2\cos^2(2x)-1$ Is this useful at all?

A:

Yes, $\cos4x=2\cos^2(2x)-1$ is useful, also $\cos^2x=\dfrac{1+\cos2x}{2}$.
Now
$$4\cos^4x-2\cos2x-1/2\cos4x=4(\dfrac{1+\cos2x}{2})^2-2\cos2x-1/2(2\cos^2(2x)-1)$$
then let $\cos2x=k$. Can you proceed?

