Question

Which of the following is the total general solution of the equation $2^{\frac{1}{1-|\cos x|}}=2^2$ ?

• A. $x=\{n\pi \pm \frac{\pi }{6},n\in Z\}$
• B. $x=\{n\pi \pm \frac{\pi }{3},n\in Z\}$
• C. $x=\{n\pi \pm \frac{\pi }{4},n\in Z\}$
• D. All of the above

Answer 1

The correct option is B $x=\{n\pi \pm \frac{\pi }{3},n\in Z\}$

$2^{\frac{1}{1-|\cos x|}}=2^2$
By comparing powers, we get
$\Rightarrow \frac{1}{1-|\cos x|}=2$
$\Rightarrow |\cos x|=\frac{1}{2}$

$\Rightarrow {\cos }^{2}x=(\frac{1}{2}{)}^2$

$\Rightarrow {\cos }^{2}x={\cos }^2\frac{\pi }{3}$
Using
${\cos }^2\theta ={\cos }^2\alpha \Rightarrow \theta =n\pi \pm \alpha ,n\in Z$
$\Rightarrow x=n\pi \pm \frac{\pi }{3},n\in Z$