Question

The most general solution of $2^{1+\left|\cos x\right|+{\cos }^{2}x+{\left|\cos x\right|}^3+\cdots \infty }=4$ are given by

• A. $k\pi \pm \frac{\pi }{3}$
• B. $2k\pi \pm \frac{\pi }{3}$
• C. $2k\pi \pm \frac{2\pi }{3}$
• D. $(2k\pm l)\frac{\pi }{2}$ for all cases $k\in l$

Answer 1

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

Given, $2^{1+\left|cosx\right|+\left|{\cos }^{2}x\right|+\left|{\cos }^{3}x\right|+....}=2^2$
$\Rightarrow 2^{\frac{1}{1-\left|cosx\right|}}=2^2\Rightarrow \frac{1}{1-\left|cosx\right|}=2$
$\Rightarrow \left|\cos x\right|=\frac{1}{2}\Rightarrow \cos x=\pm \frac{1}{2}$
$\therefore x=\frac{\pi }{3},\frac{2\pi }{3},\frac{-\pi }{3},\frac{-2\pi }{3}(\because x\in (-\pi ,\pi ))$
Thus, the solution set is $\{\pm \frac{\pi }{3},\pm \frac{2\pi }{3}\}$