Why doesn't ArcTan[Tan[a]] get "a"？

Questions
1

Hello,

In Mathematica,

ArcTan[Tan[Pi/6]]

will produce Pi/6.

But

FullSimplify@ArcTan[Tan[\[Theta]]]

only produces

ArcTan[Tan[\[Theta]]]

What is the problem of the latter?

2

Try

ArcTan[Tan[Pi/6 + Pi]]
3

May you tell why this works?

4

The equality

ArcTan[Tan[\[Theta]]] == \[Theta]

is true only when -Pi/2 < \[Theta] < Pi/2:

Simplify[ArcTan[Tan[\[Theta]]],
 -Pi/2 < \[Theta] < Pi/2]