AP EAMCET · Maths · Trigonometric Equations
The values of \(x\) in \((-\pi, \pi)\) which satisfy the equation \(8^{1+\cos ^2 x+\cos ^4 x+\ldots}=4^3\) are
- A \(\pm \frac{\pi}{4}, \pm \frac{3 \pi}{4}\)
- B \(\pm \frac{\pi}{6}, \frac{\pi}{3}\)
- C \(\pm \frac{\pi}{8}\)
- D \(\frac{\pi}{3}\)
Answer & Solution
Correct Answer
(A) \(\pm \frac{\pi}{4}, \pm \frac{3 \pi}{4}\)
Step-by-step Solution
Detailed explanation
\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…
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