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So I am a bit confused because I am using the "addition" identity for $\cot (\theta)$ but it's undefined. Here is the problem:

If $$f(\theta)= \cot(\theta)=-2$$ find $$f(\theta + \pi)$$

The only other way I can think of to solve it is by placing $\pi$ before $-2$. Would that result in a defined answer? If so, how do you know when to switch them? Thank you!

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    I don't really understand your question. The function $\cot$ is periodic with period $ \pi$. So if $\cot \theta=y$ has solution $\theta = x $ then $x+n\pi$ is also a solution for any whole number $n$. – 2017-02-02
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    Thanks Doug I did what you said I drew a unit circle. And cot theta would be -2 circa 270 degrees correct? Then if I go clockwise 180 degrees that leaves me with the point (0,1) and it would be zero at that point but now what? – 2017-02-02

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Draw your unit circle.
Find a values of $\theta$, where $cot\theta$ equal $-2.$
You don't have to find it exactly, an approximation will do. Now go around the circle $\pi$ radians what is the value of $\cot \theta$ at that point?

Next thought: $\cot \theta = \frac {\cos\theta} {\sin \theta}.$
Do you have an identity of what $\cos(\theta + \pi)$ might be?
How about $\sin (\theta + \pi)$?