Trigonometric equations involving $\prod_{r=1}^{n}\cos\left(\frac{x}{2^n}\right)$

Question

It is a well known fact in trigonometry that, for $0 < x < \pi$, we have
$\displaystyle \prod_{r=1}^{n}\cos\left(\frac{x}{2^r}\right) = \frac{\sin x}{2^n \sin\left(\frac{x}{2^n}\right)}$. This result proves useful when dealing with limits of certain functions, or summing trigonometric series. I started to wonder if there are any other useful applications of this result. Can anyone suggest any trigonometric equations that this result makes easier to solve? A pretty obvious one is $\frac{1}{8}\csc\left(\frac{x}{8}\right)=\prod_{r=1}^{3}\cos\left(\frac{x}{2^r}\right)$, but I am looking to see if there are more creative examples.

Answer 1

For you!

Solve the following equation. $$8x(2x^2-1)(8x^4-8x^2+1)=1$$