Unexpected result from evaluating ArcTan

Questions
,
1

Simplify[ArcTan[Cos[[Phi]],Sin[[Phi]]]]
I was expecting [Phi], but Wolfram doesn’t do anything about it, it just spits it out as is. How to get the desired result:
ArcTan[Cos[Phi],Sin[Phi]] = ArcTan[Tan[Phi]]=[Phi]?
Many thanks!

2

ArcTan[Tan[Phi]]=[Phi] is not true unless Phi is in the range of ArcTan.

Try

Simplify[ArcTan[Cos[\[Phi]], Sin[\[Phi]]], -Pi < \[Phi] <= Pi]
3

That solved my problem, thank you!

4

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]