To solve for $s$ in the interval $[-2\pi, \pi]$ given that $\cot^2(s) = 1$, we can proceed as follows:

1. Rewrite the Equation: The equation $\cot^2(s) = 1$ implies that $\cot(s) = \pm 1$.

2. Identify Values of $s$: The cotangent function $\cot(s)$ is equal to $\pm 1$ at specific angles:

   • $\cot(s) = 1$ at $s = \frac{\pi}{4} + k\pi$ where $k$ is an integer.
   • $\cot(s) = -1$ at $s = \frac{3\pi}{4} + k\pi$ where $k$ is an integer.

3. Find All Solutions in the Interval $[-2\pi, \pi]$:

   • For $\cot(s) = 1$:

     • $s = \frac{\pi}{4}$
     • $s = \frac{\pi}{4} - \pi = -\frac{3\pi}{4}$
     • $s = \frac{\pi}{4} - 2\pi = -\frac{7\pi}{4}$

   • For $\cot(s) = -1$:

     • $s = \frac{3\pi}{4}$
     • $s = \frac{3\pi}{4} - \pi = -\frac{\pi}{4}$
     • $s = \frac{3\pi}{4} - 2\pi = -\frac{5\pi}{4}$

4. List All Values of $s$ in the Interval:

   • The values of $s$ are: $\frac{\pi}{4}, -\frac{3\pi}{4}, -\frac{7\pi}{4}, \frac{3\pi}{4}, -\frac{\pi}{4}, -\frac{5\pi}{4}$.

So, the solution is: $s = \frac{\pi}{4}, -\frac{3\pi}{4}, -\frac{7\pi}{4}, \frac{3\pi}{4}, -\frac{\pi}{4}, -\frac{5\pi}{4}$

Would you like further details on this solution or have any questions?

Here are five related questions for further practice:

1. What is the solution for $\cot^2(x) = 1$ in the interval $[0, 2\pi]$?
2. How would you solve $\tan^2(x) = 1$ in a similar interval?
3. What is the general solution for $\cot(x) = 1$ without any interval restriction?
4. How would the answer change if the interval was $[-\pi, \pi]$ instead?
5. What are the steps to solve an equation involving $\cot^2(x)$ in any arbitrary interval?

Tip: Remember that $\cot(x)$ has a period of $\pi$, so solutions will repeat every $\pi$ interval.