Question

If $R\subset (0,\pi )$ denote the set of values of which satisfies the equation $2^{(1+\left|\cos x\right|+\left|co{s}^{2}x\right|+\left|co{s}^{3}x\right|)+\left|co{s}^{4}x\right|...............\infty }=4$ then $R$ equals

• 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 B $\{\frac{\pi }{3},\frac{2\pi }{3}\}$

$2^{(1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+.........\infty )}=2^2\Rightarrow 1+\left|\cos x\right|+\left|{\cos }^{2}x\right|+.........\infty =2\Rightarrow \frac{1}{1-\left|\cos x\right|}=2\Rightarrow 1-\left|\cos x\right|=\frac{1}{2}\Rightarrow \left|\cos x\right|=\frac{1}{2}\Rightarrow x=\frac{\pi }{3},\frac{2\pi }{3}$
in the range $(0,\pi )$