$$\sin x = 2\sin \frac{x}{2}\cos \frac{x}{2} = 4\sin \frac{x}{4}\cos \frac{x}{4}\cos \frac{x}{2} = ... = {2^n}\sin \frac{x}{{{2^n}}}\prod\limits_{k = 1}^n {\cos \frac{x}{{{2^n}}}} $$