## Precalculus (6th Edition)

$8\cos^{4}x-8\cos^{2}x+1$ (equivalent answers may differ if we choose other double angle identities for cosine)
$\cos 2A=\cos^{2}A-\sin^{2} A$ $\cos 2A=1-2\sin^{2}A$ $\cos 2A=2\cos^{2}A- 1$ -------------- (we'll use the third identity, twice) $\cos 4x=\cos[2(2x)]$ $=2\cos^{2}(2x)- 1$ $=2(\cos 2x)^{2}- 1$ $=2(2\cos^{2}x- 1)^{2}- 1$ .... $(A-B)^{2}=A^{2}-2AB+B^{2}...$ $=2(4\cos^{4}x-4\cos^{2}x+1)-1$ $=8\cos^{4}x-8\cos^{2}x+1$ (equivalent answers may differ if we choose other double angle identities for cosine)