To determine all the solutions to the equation [tex]\(\frac{1}{\cos \theta} = 2 \cos \theta\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex], let's go through the steps methodically:
1. Start with the given equation:
[tex]\[
\frac{1}{\cos \theta} = 2 \cos \theta
\][/tex]
2. Multiply both sides by [tex]\(\cos \theta\)[/tex] to clear the fraction (assuming [tex]\(\cos \theta \neq 0\)[/tex]):
[tex]\[
1 = 2 \cos^2 \theta
\][/tex]
3. Rearrange the equation:
[tex]\[
2 \cos^2 \theta = 1 \implies \cos^2 \theta = \frac{1}{2}
\][/tex]
4. Solve for [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}
\][/tex]
5. Determine the angles [tex]\(\theta\)[/tex] that satisfy [tex]\(\cos \theta = \frac{\sqrt{2}}{2}\)[/tex] and [tex]\(\cos \theta = -\frac{\sqrt{2}}{2}\)[/tex]:
- [tex]\( \cos \theta = \frac{\sqrt{2}}{2}\)[/tex] corresponds to [tex]\(\theta = \frac{\pi}{4}\)[/tex] and [tex]\(\theta = \frac{7\pi}{4}\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex].
- [tex]\( \cos \theta = -\frac{\sqrt{2}}{2}\)[/tex] corresponds to [tex]\(\theta = \frac{3\pi}{4}\)[/tex] and [tex]\(\theta = \frac{5\pi}{4}\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex].
6. Compile all solutions:
[tex]\[
\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}
\][/tex]
Thus, the complete set of solutions to the equation [tex]\(\frac{1}{\cos \theta} = 2 \cos \theta\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] is:
[tex]\[
\left\{\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}\right\}
\][/tex]
Hence, the correct choice from the given options is:
[tex]\[
\left\{\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5\pi}{4}, \frac{7 \pi}{4}\right\}
\][/tex]
