How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $?

$\cos(\alpha)$ and $\cos(\beta)$ are periodic functions.

Now assume that $\cos(\alpha\beta)=f(\cos(\alpha),\cos(\beta))$ and set $\alpha=\beta=x$. Then $\cos(x^2)=f(\cos(x),\cos(x))$ must be a periodic function, which is obviously false. So such a formula cannot exist.

Similar reasoning with $\cos((x+1)/x)$ excludes a formula for division.

This contrast with the case of addition, for which "$\cos(2x)$ must be periodic" raises no contradiction.

$\cos(n\alpha)$ is indeed expressible as a polynomial in terms of $\cos(\alpha)$ (and $\sin(\alpha)$). Reciprocally, you can in some cases solve that polynomial to obtain $\cos(\alpha/n)$ in terms of $\cos(\alpha)$.

But there is nothing like formulas expressing $\cos(\alpha\beta)$ or $\cos(\alpha/\beta)$ in terms of $\cos(\alpha)$ and $\cos(\beta)$.

Just like $e^{\alpha\beta}$ and $e^{\alpha/\beta}$ are not expressible in terms of $e^\alpha$ and $e^\beta$.