Question

The value of x in $(0,\pi )$ which satisfy the equation
$8^{-+|cosx|+co{s}^{2}x+|co{s}^{3}x|+\cdots to\infty }=4^3$ is

• A. $\{\frac{\pi }{2},\frac{3\pi }{4}\}$
• B. $\{\frac{\pi }{4},\frac{3\pi }{4}\}$
• C. $\{\frac{\pi }{3},\frac{2\pi }{3}\}$
• D. $\{\frac{\pi }{6},\frac{5\pi }{6}\}$

Answer 1

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

$8^{1+|cosx|+|cosx{|}^2+|cosx{|}^3+\cdots to\infty }=4^3[\because co{s}^{2}x=|co{s}^{2}x|also|co{s}^{n}x|=|cosx{|}^n]\Rightarrow 8^{\frac{1}{1-|cosx|}}=4^3\Rightarrow 2^{\frac{3}{1-|cosx|}}=2^6\therefore \frac{3}{1-|cosx|}=6\Rightarrow 1-|cosx|=\frac{1}{2}\therefore |cosx|=\frac{1}{2}\Rightarrow cosx=\pm \frac{1}{2}\therefore x=\frac{\pi }{3},\frac{2\pi }{3}$