Rewriting a product of cosines of multiple angles in terms of a product of cosines of a single angle

I know that

$$\sin^2(2^n x) = 4^n\sin^2(x)\left(\prod_{j=1}^n \cos(2^{n-j}x)\right)^2$$

for $n \in \mathbb{N}$.

I am trying to derive an expression for $\sin^2(2^n x)$ in terms of only sines and cosines that are functions of $x$ only (i.e. without multiple-angles).

How can I rewrite $$\prod_{j=1}^n \cos(2^{n-j}x) = \cos(x)\cos(2x)\cdots\cos(2^{n-1}x)$$ in terms of $\sin(x), \cos(x)$ only?

Thanks for any help.