International Mathematical Olympiad 1966 Problem 4

Prove that for every natural number $n$, and for every real number $x \neq k\pi / 2^t$ ($t = 0,1,\dots,n; k$ any integer) $$\frac{1}{\sin 2x} + \frac{1}{\sin 4x} + \cdots + \frac{1}{\sin 2^n x} = \cot x - \cot 2^n x.$$