#### Answer

$8\cos^{4}x-8\cos^{2}x+1$
(equivalent answers may differ
if we choose other double angle identities for cosine)

#### Work Step by Step

$\cos 2A=\cos^{2}A-\sin^{2} A$
$\cos 2A=1-2\sin^{2}A$
$\cos 2A=2\cos^{2}A- 1$
--------------
(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)