Interesting Trigonometry IMO Problem

Geometry Level 4

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

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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