Right, there is no Maclaurin expansion for for just that reason: the above series expansion for is a so called Laurent-series,nota Maclaurin series.

For one, you could determine the Maclaurin expansion for the usual way.Could I derive this cotangent representation from the series representation of sine and cosine? Or tangent? If not, how would I?

For another: much depends on what definition of the Bernoully numbers you are using. The derivation that I know of uses the "definition" of , then first derives the Maclaurin expansion for , and finally gets the Maclaurin expansion for via the identity .