Question

Find the value of x in $(-\pi ,\pi )$ which satisfy the equation $8^{1+|cosx|+co{s}^{2}x+|co{s}^{3}x|+.....}$ to infinity $=4^3$

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

Answer 1

The correct option is A $x=\pm \frac{\pi }{3},\pm \frac{2\pi }{3}$

$8^{1+|cosx|+co{s}^{2}x+|co{s}^{3}x|+.....}=4^3$
$\Rightarrow 2^{3\left(\frac{1}{1-\left|\cos x\right|}\right)}=2^6$
$\Rightarrow \frac{1}{1-\left|\cos x\right|}=2$
$\Rightarrow \left|\cos x\right|=\frac{1}{2}\Rightarrow \cos x=\pm \frac{1}{2}$

$\cos x=\frac{1}{2}\Rightarrow \cos x=\cos \frac{\pi }{3}\Rightarrow x=\pm \frac{\pi }{3}$ $[\because \cos (\theta )=\cos (-\theta \left)\right]$

$\cos x=-\frac{1}{2}\Rightarrow \cos x=\cos \frac{2\pi }{3}\Rightarrow x=\pm \frac{2\pi }{3}$
$\therefore x=\pm \frac{\pi }{3},\pm \frac{2\pi }{3}$