Question

The value of $x$ which satisfies $8^{1+\cos x+{\cos }^{2}x+....}=64$ in $[-\pi ,\pi ]$ is

• A. $\pm \frac{\pi }{2},\pm \frac{\pi }{3}$
• B. $\pm \frac{\pi }{3}$
• C. $\pm \frac{\pi }{2},\pm \frac{\pi }{6}$
• D. $\pm \frac{\pi }{6},\pm \frac{\pi }{3}$

Answer 1

The correct option is B $\pm \frac{\pi }{3}$

The series $1+\cos x+{\cos }^{2}x+...$ is an infinite G.P with common ratio $\cos x$

$8^{\frac{1}{1-\cos x}}=8^2$

$\Rightarrow \frac{1}{1-\cos x}=2$

$\Rightarrow 1=2-2\cos x$

$\Rightarrow \cos x=\frac{1}{2}$

$\Rightarrow x=\pm \frac{\pi }{3}$