How to prove $\cos(n!) \neq 1$ without using $\pi$ is irrational

If $\cos(n!) = 1$ for some $n\in\mathbb N,$ then $n! = 2\pi m$ for some $m\in\mathbb N,$ so $\pi = m/(n!)$ and thus $\pi$ is rational.

Conversely if $\pi$ is rational then $\pi = m/\ell$ for some $m,\ell\in\mathbb N,$ and $\ell$ is a divisor of $n!$ for some $n\in\mathbb N,$ so then $\cos(n!)=1.$

Therefore the only way to prove $\cos(n!)\ne1$ for every $n\in\mathbb N$ is by proving that $\pi$ is irrational. Various ways of doing that exist: https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational