$\begin{align*}
&\cos^4(x) = \\ \\
&(\cos^2(x))^2 = \\ \\
&\dfrac{(1+\cos(2x))^2}{4} = \\ \\
&\dfrac 1 4\left(\cos^2(2x)+2\cos(2x)+1\right)
\end{align*}$
From above letting $x \to 2x$
$\cos^2(2x) = \dfrac{1+\cos(4x)}{2}$
So combining these we get
$\begin{align*}
&\cos^4(x) = \\ \\
&\dfrac 1 4\left(\cos^2(2x)+2\cos(2x)+1\right) = \\ \\
&\dfrac 1 4\left(\dfrac{1+\cos(4x)}{2}+2\cos(2x) + 1\right)
\end{align*}$
and you can complete the remaining algebra