Why do graph plotting applications draw $\frac{1}{\tan x}$ the same as $\frac{\cos x}{\sin x}$?

Question

At $x=\frac{\pi}{2}$ desmos acts like they both have an intersection with the line $x=\frac{\pi}{2}$, even though obviously the function $\frac{1}{\tan x}$ is undefined at $\frac{\pi}{2}$.

Why is that?

Answer 1

In desmos, there are three types of "undefineds:"

1. Real undefined. This undefined is what we usually mean by the notion of infinity. Desmos has two "real infinities:" positive and negative. However, as a whole, both the positive real and the negative real infinities correspond to the same type.

2. Imaginary undefined. This undefined represents pretty much everything besides the real infinities. The most common way to get them is by taking the square root of a negative number. However, there are many other ways, such as going over/under the index of a list.

3. Inequality undefined. This undefined crops up when you use inequalities, namely restricting functions using curly brackets ({ and })

You con see how these three undefineds act in this graph.

When you plug in $x = \frac{\pi}{2}$ into $\tan(x)$, you get the first infinity, the real infinity. And because the reciprocal of the real infinity is $0$, $\frac{1}{\tan(x)}$ is zero for $x = \frac{\pi}{2}$, and has nothing to do with $\frac{\cos(x)}{\sin(x)}$.