The expression \(\arctan(\tan(x))\) simplifies based on the value of \(x\). Specifically, it yields:

\[
\arctan(\tan(x)) = x \quad \text{if} \quad x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)
\]

However, for values of \(x\) outside this interval, \(\arctan(\tan(x))\) will equal \(x\) modulo \(\pi\) (i.e., it will return a value within the principal range of the \(\arctan\) function).

To summarize:
- For \(x\) in \((- \frac{\pi}{2}, \frac{\pi}{2})\), \(\arctan(\tan(x)) = x\).
- For \(x\) outside this range, it adjusts to fall within the principal range by subtracting or adding \(\pi\).

Would you like more details on this topic, or do you have any specific questions? Here are some related questions to consider:

1. What is the definition of the tangent function?
2. How do the ranges of trigonometric functions affect their inverses?
3. What is the graphical representation of \(\tan(x)\) and \(\arctan(x)\)?
4. Can you explain how periodicity affects the inverse trigonometric functions?
5. How do you find the value of \(\tan(x)\) for specific angles? 

Tip: Remember that inverse trigonometric functions are defined over specific ranges to ensure they produce unique outputs.

Explore the simplification of arctan(tan(x)), including its behavior within and outside the interval (-π/2, π/2). This detailed explanation covers key concepts in trigonometry and inverse functions, suitable for Grades 10-12 students.

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