The important point is that “identity” is true only when .
In another word, it is not true when the terminal arm falls in QII and QIII.
Note that the angle lies in is also an acute angle, because the minus sign only refers to moving the terminal arm in the clockwise direction. The resultant angle is still acute.
Added: To move the terminal arm through in the anticlockwise direction is the same as moving it through in the clockwise direction.