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

1 Answer

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