The way I have understood, $0/0$ is undefined or indeterminate because, if $c=0/0$ then $c\cdot 0=0$, where $c$ can be any finite number including $0$ itself.
If we also observe a fraction $F=a/b$ where $a,b$ are positive real numbers, the value of $F$ increases with the decrements of $b$. Being $0$ the least non-negative integer, if $b$ tends $0$ then $F$ tends to $\infty$ which is greater than all finite numbers.
I have also heard that no number is equal to infinity, a variable can tend to infinity. Any ratio is also a variable or is resolved to a number.
Now,
$$\tan(\pi/2) = \frac{\sin(\pi/2)}{\cos(\pi/2)}=\frac{1}{0},$$
is it undefined or infinity?
Also we know
$$\tan2x= \frac{2\tan x}{1-\tan^2x}$$
Is this formula valid for $x=\pi/2$?
Any rectification is more than welcome.
Some references: