Q63 / 160 · AP EAMCET 2024 (22 May Shift 2) · Mathematics

The values of $x$ in $(-\pi, \pi)$ which satisfy the equation $8^{1+\cos ^2 x+\cos ^4 x+\ldots}=4^3$ are

  • Correct answer
    A
    $\pm \frac{\pi}{4}, \pm \frac{3 \pi}{4}$
  • B
    $\pm \frac{\pi}{6}, \frac{\pi}{3}$
  • C
    $\pm \frac{\pi}{8}$
  • D
    $\frac{\pi}{3}$

Solution

$\begin{aligned} & 8^{1+\cos ^2 x+\cos ^4 x+\ldots}=4^3 \Rightarrow 2^{3\left(\frac{1}{1-\cos ^2 x}\right)}=2^6 \\ \Rightarrow & \frac{3}{\sin ^2 x}=6 \Rightarrow \sin ^2 x=\frac{1}{2} \\ \Rightarrow & \sin x= \pm \frac{1}{\sqrt{2}} \Rightarrow x= \pm \frac{\pi}{4}, \pm \frac{3 \pi}{4}\end{aligned}$

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

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