Question

Let $S\in (0,\pi )$ denote the set of values of $x$ satisfying the equation $1+|\cos x|+{\cos }^{2}x+|{\cos }^{3}x|+...\infty =2$, then, $S=$

Answer 1

$1+|\cos x|+{\cos }^{2}x+|{\cos }^{3}x|....+\infty =2$

$\therefore$ Above is G.P.

With $a=1$

$r=|\cos x|$

$\therefore sum=\frac{a}{1-r}$

$\therefore \frac{1}{1-|\cos x|}=2$

$\therefore 1-|\cos x|=\frac{1}{2}$

$\therefore |\cos x|=\frac{1}{2}$

$\therefore \cos x=\pm \frac{1}{2}$

$\therefore x=\frac{\pi }{3}$

$x=\frac{2\pi }{3}$

Set of value of x is ($\frac{\pi }{3},\frac{2\pi }{3}$).