Question

Let $S\subset (0,\pi )$ denotes the set of values of $x$ satisfying the equation $8^{1+|\cos x|+{\cos }^{2}x+|{\cos }^{3}x|+\cdots +\infty }=4^3$. Then, $S=$

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

Answer 1

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

We know that,
$|\cos x|<1\forall x\in (0,\pi )$
So,
$1+|\cos x|+{\cos }^{2}x+|{\cos }^{3}x|+\cdots +\infty =\frac{1}{1-|\cos x|}$
Therefore,
$8^{1+|\cos x|+{\cos }^{2}x+|{\cos }^{3}x|+\cdots +\infty }=4^3\Rightarrow 2^{3(1+|\cos x|+{\cos }^{2}x+|{\cos }^{3}x|+\cdots +\infty )}=2^6\Rightarrow \frac{1}{1-|\cos x|}=2\Rightarrow |\cos x|=\frac{1}{2}\Rightarrow \cos x=\pm \frac{1}{2}\Rightarrow x=\frac{\pi }{3},\frac{2\pi }{3}\Rightarrow S=\{\frac{\pi }{3},\frac{2\pi }{3}\}$