A "good enough" computer algebra system should be able to handle
computations with IND INF UND (etc) at least as well as a
fully-implemented IEEE float system, and probably a good
deal more effectively.
There have been a number of proposals to do so, including tracking the
sources of infinity, so that
INF-INF is 0 IF THEY ARE THE SAME INF.
but INF-INF is UND otherwise..
as for tan(pi/2), consider 3+1/tan(pi/2).
Can you agree on a value for that?
RJF