Explanation

(b)
Now, $1 + |\cos{x| + \cos^{2}{x + |\cos^{3}{x| + ...}\infty = \frac{1}{1 - |\cos{x|}}}}$
$\therefore\ \ \frac{1}{8^{1 - |\cos{x|}}} = 4^{3}$
$\Rightarrow \ \frac{3}{2^{1 - |\cos{x|}}} = 2^{6}\ \Rightarrow \ 1 = 2 - 2|\cos{x|}$
$\Rightarrow |\cos{x| = \frac{1}{2}}$
$\Rightarrow \cos{x = \pm \frac{1}{2}}$
$\Rightarrow x = \frac{\pi}{3}, - \frac{\pi}{3},\frac{2\pi}{3},\ - \frac{2\pi}{3}$
$\therefore$ Number of solutions $= \ 4$