With the beta v2, I tried:
tsimplify(atan(cot(pi*x)))
which returned -pi*x-(1/2)*pi
...which isn't correct to the best of my knowledge (doesn't plot the same).
tsimplify replaces trig expressions with exp and inverse trig with ln then it does simplifications without taking care of complex ln branches, the result is not necessarily valid on the whole complex plane.
That doesn't look like what it did here...???
I mean the results are not equivalent to the input function.
Perhaps Parisse should not have said "complex" because (unlike e.g. exp) tan is already periodic on the real line, and so atan has real branches there, separated by multiples of pi.
So the "simplified" answer is correct, up to a multiple pi, which in my opinion is probably as much as you should expect when you are using variable arguments (as opposed to an explicit numerical argument that would allow you to obtain a "principal value").
It's actually valid for x in [-1,0], and y in [-1.5,1.5] approx.
-Donald