Series $\sum_{k=1}^n\frac{\tan\frac x{2^k}}{2^{k-1} \cos\frac x{2^{k-1}}}$ Evaluate
$$\sum_{k=1}^n\frac{\tan\frac x{2^k}}{2^{k-1} \cos\frac x{2^{k-1}}}$$
I am struck with calculation for this question

Hint:$$2\cot (2x)=\cot x-\tan x\\ \to \tan x=2\cot (2x)-\cot x$$so 
$$\frac{\tan\frac x{2^k}}{2^{k-1} \cos\frac x{2^{k-1}}}=\\
\frac{2\cot (2\frac x{2^k})-\cot \frac x{2^k}}{2^{k-1} \cos\frac x{2^{k-1}}}$$
