Step 1: Understanding the Concept:
We need to find the general solution for a trigonometric equation involving the square of the tangent function. This requires using the standard formula for general solutions of this type.
Step 2: Key Formula or Approach:
The general solution for an equation of the form $\tan^2 x = \tan^2 \alpha$ is given by:
\[ x = n\pi \pm \alpha \]
where n is any integer.
Step 3: Detailed Explanation:
The given equation is:
\[ \tan^2 x = 1 \]
We need to find an angle $\alpha$ such that $\tan^2 \alpha = 1$. We know that $\tan(\frac{\pi}{4}) = 1$.
Therefore, $\tan^2(\frac{\pi}{4}) = (1)^2 = 1$.
So, we can write the equation as:
\[ \tan^2 x = \tan^2\left(\frac{\pi}{4}\right) \]
This is now in the form $\tan^2 x = \tan^2 \alpha$, with $\alpha = \frac{\pi}{4}$.
Using the general solution formula from Step 2:
\[ x = n\pi \pm \frac{\pi}{4} \]
where n is any integer.
Step 4: Final Answer:
The general solution for the given equation is $x = n\pi \pm \frac{\pi}{4}$. Therefore, option (B) is correct.