S_n=\sum^{n}_{k=1}lncos\frac{a}{2^k}=ln\prod_{k=1}^ncos\frac{a}{2^k}=ln[(cos\frac{a}{2}cos\frac{a}{2^2}cos\frac{a}{2^3}...cos\frac{a}{2^{n-1}}cos\frac{a}{2^n})\cdot \frac{2sin\frac{a}{2^n}}{2sin\frac{a}{2^n}}]=