It is a little disappointing that Mathematica cannot decide that this is True:
FullSimplify[\[Phi] - 2 Pi == ArcTan[Cos[\[Phi]], Sin[\[Phi]]],
Pi < \[Phi] < 3 Pi]
The result of this is bizarre:
Reduce[\[Phi] - 2 Pi ==
ArcTan[Cos[\[Phi]], Sin[\[Phi]]] &&
Pi < \[Phi] < 3 Pi,
\[Phi], Reals]
