To find the solutions of the equation [tex]\(\tan^2 x + \sec x = 1\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex], we proceed as follows:
1. Understand the equation: We start with the trigonometric identity [tex]\(\tan^2 x + \sec x = 1\)[/tex].
2. Isolate terms if possible: Look for potential values of [tex]\(x\)[/tex] that might simplify the equation. In this case, we will substitute specific values within the interval and check if they satisfy the equation.
3. Evaluate at specific points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
\tan^2(0) + \sec(0) = 0 + 1 = 1
\][/tex]
This is a solution.
- When [tex]\( x = \frac{2\pi}{3} \)[/tex]:
[tex]\[
\tan^2\left(\frac{2\pi}{3}\right) = \tan^2\left(\pi - \frac{\pi}{3}\right) = \tan^2 \left(\frac{\pi}{3}\right) = \left(\sqrt{3}\right)^2 = 3
\][/tex]
[tex]\[
\sec\left(\frac{2\pi}{3}\right) = \sec\left(\pi - \frac{\pi}{3}\right) = -\sec\left(\frac{\pi}{3}\right) = -2
\][/tex]
[tex]\[
\tan^2\left(\frac{2\pi}{3}\right) + \sec\left(\frac{2\pi}{3}\right) = 3 - 2 = 1
\][/tex]
[tex]\(\frac{2\pi}{3}\)[/tex] is a solution.
- When [tex]\( x = \frac{4\pi}{3} \)[/tex]:
[tex]\[
\tan^2\left(\frac{4\pi}{3}\right) = \tan^2\left(\pi + \frac{\pi}{3}\right) = \tan^2 \left(\frac{\pi}{3}\right) = \left(\sqrt{3}\right)^2 = 3
\][/tex]
[tex]\[
\sec\left(\frac{4\pi}{3}\right) = \sec\left(\pi + \frac{\pi}{3}\right) = -\sec\left(\frac{\pi}{3}\right) = -2
\][/tex]
[tex]\[
\tan^2\left(\frac{4\pi}{3}\right) + \sec\left(\frac{4\pi}{3}\right) = 3 - 2 = 1
\][/tex]
[tex]\(\frac{4\pi}{3}\)[/tex] is a solution.
4. Summarize the solutions: From these evaluations, we see the valid solutions are [tex]\( x = 0 \)[/tex] and [tex]\( x = \frac{2\pi}{3} \)[/tex].
Thus, the correct set of solutions within the interval [tex]\([0, 2\pi)\)[/tex] is [tex]\(\boxed{\frac{2 \pi}{3}, 0}\)[/tex]. Since [tex]\(\frac{4\pi}{3}\)[/tex] is also a valid solution, combining these results, we get the final list of solutions:
[tex]\[0, \frac{2 \pi}{3}\][/tex]
This matches the set of solutions [tex]\([0, \frac{2 \pi}{3}]\)[/tex].
Therefore, the correct choice from the given options is:
[tex]\[
\boxed{\frac{2\pi}{3}, \frac{4\pi}{3}, 0}
\][/tex]
