To solve for $\tan(t - \pi)$, we can use the trigonometric identity for the tangent function:

$\tan(t - \pi) = \tan(t)$

This identity tells us that subtracting $\pi$ from the angle $t$ does not change the sign or the value of the tangent function. Therefore:

$\tan(t - \pi) = \tan(t)$

Given that $\tan(t) = 49$, we have:

$\tan(t - \pi) = 49$

So, the exact value of $\tan(t - \pi)$ is $49$.

Would you like more details, or do you have any other questions?

Here are some related questions:

1. What is the general identity for $\tan(t + \pi)$?
2. How does the tangent function behave when the angle is shifted by $\frac{\pi}{2}$?
3. What is the period of the tangent function?
4. What is the value of $\tan\left(\frac{\pi}{4}\right)$?
5. How can the tangent function be derived using sine and cosine?
6. What happens to $\tan(t)$ if $t$ is shifted by $\frac{\pi}{3}$?
7. How is the tangent function related to the unit circle?
8. What is the relationship between $\tan(t)$ and $\cot(t)$?

Tip: Remember that the tangent function has a period of $\pi$, meaning $\tan(t + \pi) = \tan(t)$.