<s>[INST] 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?
 [/INST] Hint: using $\cos2x = 1-2\sin^2(x)$, $\sin(2x) = 2\sin(x)\cos(x)$ and $\sin^2(x) + \cos^2(x) = 1$ to get the following and simplify it:
$$S = 4\cos^4(x)-2(1-2sin^2(x))+1/2(1-2sin^2(2x))$$
$$S = 4\cos^4(x)-2+4\sin^2(x)+1/2 - 4sin^2(x)cos^2(x)$$
$$S = 4\cos^4(x)-2+1/2 + 4\sin^2(x)(1-\cos^2(x)) = 4\cos^4(x)-2+1/2 + 4\sin^4(x) = 4 - 2 + 1/2 = 2.5$$
I should notice that I think one of the $-$ is $+$ (in my case the second minus).
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