NL-NWO 1988 Problem 2

Given is a number $a$ with 0 $\le \alpha \le \pi$ . A sequence $c_0,c_1, c_2,...$ is defined as $$c_0=\cos \alpha$$ $$C_{n+1}=\sqrt{\frac{1+c_n}{2}} \, for \,\, n=0,1,2,...$$ Calculate $\lim_{n\to \infty}2^{2n+1}(1-c_n)$