The more "popular" function is a continuous ski-slope through y=pi/2 at x=0. It can be found in books, all the tutoring web sites, and other locations. The most prominent graph can be found at: http://en.wikipedia.org/wiki/Inverse_trigonometric_functions It is clear from the graph that this arccot is complementary (sums to a constant value) to the arctan function.
I understand the inverse trig functions are multi-valued and their principal values depend on somewhat arbitrary branch cuts. I suspect the differences come from the way the functions are derived--directly from the cotangent function or indirectly from another inverse function. (I have read one article that states arccot is defined as the complement of arctan.)
Any light anyone can shed on this topic will be appreciated. Thanks.