The most general solution of

$2^{1+|\cos x|+{\cos }^{2}x+\left|{\cos }^{3}x\right|+\dots =4}$  is given by

The given equation is not meaningful for 

$|\cos x|=1\text{. So, let }|\cos x|\ne 1.$

Now,

   $2^{1+|\cos x|+{\cos }^{2}x+|\cos x{|}^3+\dots +\mathrm{tos}x}=4$

$\begin{array}{l}\Rightarrow \frac{1}{2^{1-|\cos x|}}=2^2\\ \Rightarrow \frac{1}{1-|\cos x|}=2\\ \Rightarrow 2-2|\cos x|=1\\ \Rightarrow |\cos x|=\frac{1}{2}\\ \Rightarrow \cos x=\pm \frac{1}{2}\\ \Rightarrow \cos x=\cos \frac{\pi }{3},\cos x=\cos \frac{2\pi }{3}\\ \Rightarrow x=2n\pi \pm \frac{\pi }{3},x=2n\pi \pm \frac{2\pi }{3},n\in Z\\ \Rightarrow x=2n\pi \pm \frac{\pi }{3},x=(2n\pm 1)\pi \pm \frac{\pi }{3}\\ \Rightarrow x=n\pi \pm \frac{\pi }{3},n\in Z.\end{array}$