Why is tan^-1(tan 10pi/11) = -pi/11 ?

Questions
1

I thought that if the inverse was taken to its function, it would just cancel itself out and leave the value inside.
Thus, tan^-1(tan 10pi/11) = 10pi/11,

But the correct answer is -pi/11. Can someone please explain to me why?

I would greatly appreciate the help.
Thank you.

2

Tan is not invertible as a whole. ArcTan is the inverse of the restriction to the interval between -Pi/2 to Pi/2.
Your angle 10Pi/11 is outside that interval. The angle between -Pi/2 and Pi/2 which has the same Tan as 10Pi/11 is precisely -Pi/11.
Have a look:

With[{\[Theta] = 10 Pi/11}, 
 Graphics[{Circle[], PointSize[Large], 
   Point[{Cos[\[Theta]], Sin[\[Theta]]}], Line[{{1, -1}, {1, 1}}], 
   Line[{{Cos[\[Theta]], Sin[\[Theta]]}, 
     2 {Cos[\[Theta] - Pi], Sin[\[Theta] - Pi]}}]}, Axes -> True]]