We are given that:

$\text{tan}(t) = \frac{4}{9}$

We need to find the value of $\text{tan}(t - \pi)$.

Using the periodicity property of the tangent function, we know that:

$\text{tan}(t - \pi) = \text{tan}(t)$

This property comes from the fact that the tangent function has a period of $\pi$, meaning $\text{tan}(t + \pi) = \text{tan}(t)$.

Therefore, the exact value of $\text{tan}(t - \pi)$ is:

$\boxed{\frac{4}{9}}$

Would you like further explanation on the periodic properties of the tangent function?

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Here are 5 related questions for further exploration:

1. What is the general formula for the period of a trigonometric function?
2. If $\tan(t) = \frac{4}{9}$, what is $\tan(t + \frac{\pi}{2})$?
3. How does the tangent function behave for angles in different quadrants?
4. What is the value of $\tan(t + 2\pi)$ based on periodicity?
5. How does the cotangent function relate to the tangent function?

Tip: Tangent and cotangent functions repeat every $\pi$ radians, unlike sine and cosine which repeat every $2\pi$.