The number of values of x lying in the internal $(-\mathrm{\pi },\mathrm{\pi })$ which satisfy the equation $8^{\left(1+|\cos \mathrm{x}|+{\cos }^2\mathrm{x}+\left|{\cos }^3\mathrm{x}\right|+\dots \mathrm{\infty }\right)}=4^3,$is 

Now,$1+|\cos \mathrm{x}|+{\cos }^2\mathrm{x}+\left|{\cos }^3\mathrm{x}\right|+\dots \mathrm{\infty }=\frac{1}{1-|\cos \mathrm{x}|}$

$\therefore 8^{\frac{1}{1-|\cos \mathrm{x}|}}=4^3$

$\begin{array}{ll}\Rightarrow & \frac{3}{1-|\cos \mathrm{x}|}=2^6\Rightarrow 1=2-2|\cos \mathrm{x}|\\ \Rightarrow & |\cos \mathrm{x}|=\frac{1}{2}\\ \Rightarrow & \cos \mathrm{x}=\pm \frac{1}{2}\\ \Rightarrow & \mathrm{x}=\frac{\mathrm{\pi }}{3},-\frac{\mathrm{\pi }}{3},\frac{2\mathrm{\pi }}{3},-\frac{2\mathrm{\pi }}{3}\end{array}$

$\therefore \text{ Number of solutions }=4$