Make many "Morrie's Laws"

Geometry Level 5

Richard Feynman was fascinated by the equation \(\cos(20^\circ)\cos(40^\circ)\cos(80^\circ)=\dfrac1{2^3}\), which he named "Morrie's Law" after a childhood friend, Morrie Jacobs, who told him the rule. More generally, how many positive integers \(n<90\) are there such that \(\displaystyle \prod_{k=0}^{m-1}\cos(2^kn^\circ)=\dfrac{1}{2^m}\) for some positive integer \(m\)?

Inspiration: here and here.
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