The problem asks you to evaluate the expression:

$\tan^{-1} \left( \tan \left( \frac{\pi}{4} \right) \right)$

Step-by-Step Solution:

1. Evaluate $\tan\left(\frac{\pi}{4}\right)$:

   • $\tan\left(\frac{\pi}{4}\right) = 1$. (Since the tangent of $\frac{\pi}{4}$ radians or 45 degrees is 1.)

2. Now the expression becomes: $\tan^{-1}(1)$

3. Find the value of $\tan^{-1}(1)$:

   • This asks, "For what angle $\theta$ is $\tan(\theta) = 1$?"
   • The answer is $\theta = \frac{\pi}{4}$, since the principal value (range) of $\tan^{-1}$ lies between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

4. Final Answer: $\tan^{-1}\left(\tan\left(\frac{\pi}{4}\right)\right) = \frac{\pi}{4}$

So the simplified answer is:

$\boxed{\frac{\pi}{4}}$

Would you like any further clarifications or have follow-up questions?

5 Related Questions:

1. What is the principal range of the inverse tangent function, $\tan^{-1}(x)$?
2. What happens if the angle inside the tangent function exceeds the principal range?
3. How would the answer change if the angle was $\frac{5\pi}{4}$ instead of $\frac{\pi}{4}$?
4. What is the relationship between the inverse functions $\tan^{-1}$ and $\tan$?
5. Can you express the tangent function in terms of sine and cosine?

Tip: Always ensure that the input angle falls within the principal range of the inverse trigonometric function to get the correct result!