IMO 1966 Problem 4

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