I have been told that $\lim_{m\to \infty}\left(\cos (x/2) + \cos (x/2^2) + \cdots + \cos (x/2^m)\right)=\frac{x}{\sin (x)}$
I tried working with the power series of $\sin(x)$ and $\cos(x)$ but it becomes messy pretty soon. I was hoping that there might be some trick to arrive at such a conclusion. I would be grateful for any suggestions.