Question

The solution of the equation $(8{)}^{1+|cosx|+|cosx{|}^2+|cosx{|}^3+...}=4^3$ in the interval $(-\pi ,\pi )$ are.

• A. $\pm \frac{\pi }{3},\pm \frac{\pi }{6}$
• B. $\pm \frac{\pi }{3},\pm \pi$
• C. $\pm \frac{\pi }{3},\pm \frac{2\pi }{3}$
• D. none of these

Answer 1

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

$(8{)}^{1+|cosx|+|cosx{|}^2+|cosx{|}^3+...}=4^3$
$\Rightarrow (8{)}^{\frac{1}{1-|\cos x|}}=4^3=64=8^2$,

since $|\cos x|<1$ in $(-\pi ,\pi )$
$\Rightarrow \frac{1}{1-|\cos x|}=2$
$\Rightarrow |\cos x|=\frac{1}{2}$
The solution in the given interval is,
$x=\pm \frac{\pi }{3},\pm \frac{2\pi }{3}$
Hence, option 'C' is correct.