Emily was asked to make a cake for a math-themed birthday party. The cake was decorated with mathematical symbols, and the guests were math enthusiasts.  
 
As part of the entertainment, the party organizer set up a math quiz for the guests. One of the questions was about trigonometry, and Emily was asked to help grade the quiz. The question was: 

If tant=49tant=49, what is tan(t−π)tant−π?
 
Help Emily find the correct answer to this question.

Enter the exact answer.

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 \(\tan(t + \pi) = \tan(t)\).

Explore how to solve \( \tan(t - \pi) \) using trigonometric identities. This problem involves understanding how trigonometric functions behave when angles are adjusted. Suitable for advanced high school students or anyone interested in trigonometry.

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