Interesting Trigonometry IMO Problem

Geometry Level 5

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

Let nn be a positive integer and xx be a real number where xkπ2mx \ne \dfrac {k \pi}{2^m} for non-negative integer mm and kk is an integer. What is the simplest form of the expression above?

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