We must know that [tex]\cos(2x)=2\cos^2(x)-1[/tex]. So:
[tex]\cos(4t)=\cos(2\cdot2t)\\\\
\cos(4t)=2\cos^2(2t)-1\\\\
\cos(4t)=2(2\cos^2(t)-1)^2-1\\\\
\cos(4t)=2(4\cos^4(t)-4\cos^2(t)+1)-1\\\\
\cos(4t)=8\cos^4(t)-8\cos^2(t)+2-1\\\\
\boxed{\cos(4t)=8\cos^4(t)-8\cos^2(t)+1}[/tex]
