$$(\cot x - 2\cot 2x)+(2\cot2x-2^2\cot 2^2x)+\cdots+(2^n\cot 2^nx - 2^{n+1}\cot 2^{n+1}x)$$ Let's evaluate each of these term: \begin{align*} &\cot x - 2\cot2x\\ &=\frac{\cos x}{\sin x}-\frac{2\cos 2x}{\sin2x}\\ &=\frac{\cos x}{\sin x}-\frac{2(\cos^2 x - \sin^2x)}{2\sin x\cos x}\\ &=\frac{\cos^2 x}{\sin x\cos x} -\frac{\cos^2 x - \sin^2 x}{\sin x\cos x}\\ &=\frac{\sin^2 x}{\sin x \cos x}\\ &=\tan x \end{align*} Therefore, we have \begin{align*} \tan x &= \cot x -2\cot 2x\\ 2\tan 2x &= 2\cot 2x - 2^2 \cot 2^2 x\\ \cdots &= \cdots \\ 2^n\tan 2^n x &= 2^n\cot 2^n x - 2^{n+1}\cot 2^{n+1}x \end{align*} Adding these $n$ equations gives the desired result.