why tan(90)=inf

It doesn't = "infinity"   [ this is what many calculators show ]

It would be better said that it is "undefined"

The tangent ratio is  y / x

But, at 90°, y will = r, but x = 0......and since we cannot dvide by 0, then thr tangent is undefined at 90°

Here is the graph of y=tanx

You can see that as x tends to 90 degrees from below   the tan of it approaches +infinity.

AND

as x tends to 90 from above x tends to - infinity.

So it is correct to say this

\(\displaystyle\lim_{x\rightarrow \infty^-}\;\;tanx=+\infty\\~\\ \displaystyle\lim_{x\rightarrow \infty^+}\;\;tanx=-\infty\\~\\ but\\ \displaystyle\lim_{x\rightarrow \infty}\;\;tanx=\text{undefined}\)