Evaluate each expression without using a calculator.
tan⁻¹ (tan (π/4))

Verified step by step guidance

1. Recall that the function \( \tan^{-1}(x) \), also known as \( \arctan(x) \), is the inverse of the tangent function \( \tan(x) \) but its output (range) is restricted to \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
2. Identify the input inside the inverse tangent: \( \tan\left(\frac{\pi}{4}\right) \). Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), the expression becomes \( \tan^{-1}(1) \).
3. Now, evaluate \( \tan^{-1}(1) \) by finding the angle \( \theta \) in the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \) such that \( \tan(\theta) = 1 \).
4. Recall that \( \tan\left(\frac{\pi}{4}\right) = 1 \), and since \( \frac{\pi}{4} \) lies within the principal range of \( \arctan \), the value of \( \tan^{-1}(1) \) is \( \frac{\pi}{4} \).
5. Therefore, the expression \( \tan^{-1}(\tan(\frac{\pi}{4})) \) simplifies to \( \frac{\pi}{4} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like tan⁻¹ (arctan), reverse the effect of their corresponding trigonometric functions. For example, tan⁻¹(x) returns the angle whose tangent is x, typically within a principal value range to ensure a unique output.

Principal Value Range of arctan

The principal value of arctan is the interval (-π/2, π/2), meaning arctan returns angles only within this range. This restriction ensures the inverse function is well-defined and single-valued, which is crucial when evaluating expressions like tan⁻¹(tan(θ)).

Periodicity and Simplification of tan and arctan

The tangent function is periodic with period π, so tan(θ) = tan(θ + nπ) for any integer n. When evaluating tan⁻¹(tan(θ)), the result is the angle equivalent to θ within the principal range of arctan, requiring adjustment if θ lies outside (-π/2, π/2).