If$x\in (-\pi ,\pi )$ such that $y=1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+\left|{\cos }^{3}x\right|+\mathrm{........}$ and $8^y=64$, then no. of values of  x is

$\begin{array}{l}{8}^y=64\\ 8^{(1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+\mathrm{.......})}=4^3\\ 2^{3(1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+\mathrm{........})}=2^6\\ \Rightarrow 3(1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+\mathrm{.....})=6\end{array}$

$\begin{array}{l}1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+\mathrm{.....}=2\\ \frac{1}{1-\left|\cos x\right|}=2\\ \left|\cos x\right|=\frac{1}{2}\\ x=\frac{\pi }{3};\frac{-\pi }{3},\frac{2\pi }{3},\frac{-2\pi }{3},\mathrm{.....}\end{array}$

The values of x is in $(-\pi ,\pi )\text{ are }\pm \frac{\pi }{3},\pm \frac{2\pi }{3}$

  $\therefore$No. of values of x is 4