Interesting Trigonometry IMO Problem

Geometry Level 5

$\large \frac{1}{\sin (2x)} + \frac{1}{\sin (4x)} + \frac1{\sin(8x) } + \ldots + \frac{1}{\sin {(2^n x)}}$

Let $$n$$ be a positive integer and $$x$$ be a real number where $$x \ne \dfrac {k \pi}{2^m}$$ for non-negative integer $$m$$ and $$k$$ is an integer. What is the simplest form of the expression above?

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