Hello Hurkyl. You got to it before I could make amends. Yea, ArcCot and ArcCos will not converge. However, I suspect the ArcSin will.Hurkyl said:A sum of reciprocals is generally not the reciprocal of the sum.
One of the advantages of the tangent function is that its sum formula involves only tangents, so the corresponding formula for arctangents is nicer.
You could try constructing similar things for the others, I suppose. Start with a telescoping series, then apply an addition formula and see what you get.