Question

The values of $x$ in $(-\pi ,\pi )$ which satisfy the equation $8^{1+|cosx|+|co{s}^{2}x|+|co{s}^{3}x|+.......\infty }=4^3$ are

• A. $\frac{\pi }{3},\frac{2\pi }{3}$
• B. $\frac{\pi }{3},\frac{-2\pi }{3}$
• C. $\frac{-\pi }{3}\frac{2\pi }{3}$
• D. $\pm \frac{\pi }{3},\pm \frac{2\pi }{3}$

Answer 1

The correct option is D $\pm \frac{\pi }{3},\pm \frac{2\pi }{3}$

$1+|\cos x|+|\cos x^2|+|{\cos }^{3}x|+.....$forms infinite G.P. with $a=1,r=|\cos x|$

using sum of GP formula we get
$8^{\frac{1}{1-|cosx|}}=64=8^2$
$|cosx|=\frac{1}{2}$
so $x=\pm \frac{\pi }{3}or\pm \frac{2\pi }{3}$