Q68 / 160 · TS EAMCET 2008 · Mathematics

$\left\{x \in R: \cos 2 x+2 \cos ^2 x=2\right\}$ is equal to

  • A
    $\left\{2 n \pi+\frac{\pi}{3}: n \in Z\right\}$
  • Correct answer
    B
    $\left\{n \pi \pm \frac{\pi}{6}: n \in Z\right\}$
  • C
    $\left\{n \pi+\frac{\pi}{3}: n \in Z\right\}$
  • D
    $\left\{2 n \pi-\frac{\pi}{3}: n \in Z\right\}$

Solution

Given equation is $$ \begin{array}{rlrl} & & \cos 2 x+2 \cos ^2 x & =2 \\ \Rightarrow & 2 \cos ^2 x-1+2 \cos ^2 x & =2 \\ \Rightarrow & 4 \cos ^2 x & =3 \\ \Rightarrow & \cos ^2 x & =\frac{3}{4} \\ \Rightarrow & & \cos x= \pm \frac{\sqrt{3}}{2} \\ & \therefore & x=n \pi \pm \frac{\pi}{6}: n \in Z \end{array} $$

Hence, the correct answer is option: 'B'.

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