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The problem presented is:

\[
\cot^2 x = 1
\]

To solve for \( x \), we can follow these steps:

1. Take the square root of both sides:

   \[
   \cot x = \pm 1
   \]

2. Now, consider the values of \( x \) for which \( \cot x = 1 \) and \( \cot x = -1 \).

   - For \( \cot x = 1 \): The value of \( x \) that satisfies \( \cot x = 1 \) within one period \( [0, \pi] \) is \( x = \frac{\pi}{4} \).
   - For \( \cot x = -1 \): The value of \( x \) that satisfies \( \cot x = -1 \) within the same interval is \( x = \frac{3\pi}{4} \).

3. Since cotangent is periodic with period \( \pi \), the general solutions are:

   \[
   x = \frac{\pi}{4} + n\pi \quad \text{and} \quad x = \frac{3\pi}{4} + n\pi
   \]

   where \( n \) is any integer.

Thus, the solutions to the equation \( \cot^2 x = 1 \) are:

\[
x = \frac{\pi}{4} + n\pi \quad \text{and} \quad x = \frac{3\pi}{4} + n\pi
\]

Would you like further details on this solution or explanations on related concepts?

Here are 5 related questions for deeper understanding:

1. What are the general properties of the cotangent function, especially its period and symmetry?
2. How does one derive the period of trigonometric functions like \( \cot x \)?
3. What is the significance of finding general solutions in trigonometric equations?
4. How would the solution change if the equation were \( \cot^2 x = 2 \) instead of \( \cot^2 x = 1 \)?
5. How can we verify the solutions of trigonometric equations graphically?

**Tip:** When solving trigonometric equations, always consider the periodic nature of the functions to find all possible solutions within the desired interval.

Learn how to solve the trigonometric equation cot^2(x) = 1 by finding values of x that satisfy cot(x) = ±1. This involves understanding the periodic properties of the cotangent function. Suitable for high school trigonometry students.

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