Inverse Trigonometric functions and their properties.

I've been recently working on Inverse Trigonometry and I've started noticing that too often there are "extra" terms such as $\pi/2$, $\pi$ or $2\pi$ that are added to certain expressions. So for instance, consider the following properties:

1. $\tan^{-1}x=\cot^{-1}x \text{ for }x>0 \text{ and } \color{red}{-\pi}+\cot^{-1}x,\text{ for } x<0.$
2. $\tan^{-1}x+\tan^{-1}y=\ \left\{ \begin{array}{ll} \tan^{-1}{\frac{x+y}{1-xy}} & \text{if }xy<1. \\ \color{red}\pi+\tan^{-1}{\frac{x+y}{1-xy}} & \text{if }x>0, y>0 \text{ and }xy>1. \\ \color{red}{-\pi}+\tan^{-1}{\frac{x+y}{1-xy}} & \text{if }x<0, y<0 \text{ and }xy>1. \end{array} \right.$

There are many more properties in which you would find a $\pi$ or a multiple of it . Now, I understand the proofs of these properties to a certain extent, but I wanted to develop an intuition about why this happens. I mean, how did mathematicians developing these functions suspect that by having a constraint on the product $xy$ we can get different values for the same expression. Also, since I'm preparing for an exam I would be glad if someone could advise me on how to go about learning these properties.

What you have to know is

$\forall x\in [-1,1]$

$$0\leq arccos(x) \leq \pi$$

$$-\frac{\pi}{2} \leq arcsin(x) \leq \frac{\pi}{2}$$

and

$\forall x \in \mathbb R$

$$-\frac{\pi}{2} <arctan(x) < \frac{\pi}{2}$$

for example

$arccos(cos(\frac{\pi}{3}+5\pi))=\frac{\pi}{3}$