Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
$\tan^{-1}\left(\tan\left(\frac{\pi}{4}\right)\right)$

Verified step by step guidance

1. Understand the problem: We need to evaluate \( \tan^{-1}(\tan(\frac{\pi}{4})) \). This involves understanding the properties of the inverse trigonometric functions.
2. Recall the definition of the inverse tangent function: \( \tan^{-1}(x) \) is the angle \( \theta \) such that \( \tan(\theta) = x \) and \( \theta \) is in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).
3. Evaluate \( \tan(\frac{\pi}{4}) \): The tangent of \( \frac{\pi}{4} \) is 1, because \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and both \( \sin(\frac{\pi}{4}) \) and \( \cos(\frac{\pi}{4}) \) are \( \frac{\sqrt{2}}{2} \).
4. Apply the inverse function: Since \( \tan(\frac{\pi}{4}) = 1 \), we have \( \tan^{-1}(1) \). The angle whose tangent is 1 and lies within \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) is \( \frac{\pi}{4} \).
5. Conclude the evaluation: Therefore, \( \tan^{-1}(\tan(\frac{\pi}{4})) = \frac{\pi}{4} \).