Solve the following equations.
$\tan^22\theta=1,0\lt{\theta}\lt{\pi}$

Verified step by step guidance

1. Recognize that the equation \( \tan^2(2\theta) = 1 \) implies \( \tan(2\theta) = \pm 1 \).
2. Consider the principal values for \( \tan(2\theta) = 1 \), which occur at \( 2\theta = \frac{\pi}{4} + n\pi \) for integer \( n \).
3. Consider the principal values for \( \tan(2\theta) = -1 \), which occur at \( 2\theta = \frac{3\pi}{4} + n\pi \) for integer \( n \).
4. Solve for \( \theta \) by dividing each equation by 2: \( \theta = \frac{\pi}{8} + \frac{n\pi}{2} \) and \( \theta = \frac{3\pi}{8} + \frac{n\pi}{2} \).
5. Determine the values of \( \theta \) that satisfy \( 0 < \theta < \pi \) by substituting different integer values for \( n \) and checking the range.