Why is $\cos(i)>1$?

It's true that the cosine of a real number must be between $-1$ and $1$, but this is not true for the cosine of a complex number. In fact, complex-differentiable functions can never be bounded (unless they are constant).

Here is an analogy, if you like. Let $f(x) = x^2$. Then we learn some rule that $f(x) \geq 0$ for all $x$. But wait a second, $f(i)$ is negative. There's no scandal, since $i$ is not a real number.

The general definition of $\cos(z)$ is $$\cos(z)=\frac{{e^{iz}}+e^{-iz}}{2}$$ When you plug in complex numbers into $\cos(z$), you can get values greater than $1$ or less than $-1$

The function $\cos z$ belongs in the interval $\left[-1,1 \right]$ when $z$ is a real number, not necessarily when $z$ is a complex number. An example of this is $\cos i > 1$, as you correctly pointed out.