Is there an alternative form/identity for $\prod_{k=1}^{m} \cos(2^kx)$? I am working with a family of integrals that involve a product of cosines of the form $\prod_{k=1}^{m} \cos(2^kx)$.  Is there a formula or identity for simplifying $\prod_{k=1}^{m} \cos(2^kx)$ to a polynomial number of terms in $x$?  Or $\cos x$, $\sin x$?

$$\cos x\cdot \cos (2x)\cdot\cos(4x)\cdot\cos(8x)...\cos(2^mx)=P$$
Multiply both sides by $\sin x$, so
$$\frac{1}{2}\sin 2x\cdot \cos (2x)\cdot\cos(4x)\cdot\cos(8x)...\cos(2^mx)=P\sin x$$
$$\frac{1}{2^2}\sin 4x\cdot\cos(4x)\cdot\cos(8x)...\cos(2^mx)=P\sin x$$
$$\frac{1}{2^3}\sin 8x\cdot\cos(8x)...\cos(2^mx)=P\sin x$$
keep doing that and get,
$$\frac{1}{2^{m+1}}\sin(2^{m+1}x)=P\cdot \sin x \Rightarrow P=\frac{\sin(2^{m+1}x)}{2^{m+1}\sin x}$$
