Let $\alpha$, $1 \lt \alpha \lt \varphi(n)$, $\gcd(\alpha, \varphi(n)) = 1$, and $\beta \equiv \alpha^{-1} \pmod {\varphi(n)}$, where $\varphi$ is Euler's totient function.
When n is even, how would I prove:
$$(n-1) \equiv (n - 1)^{\beta} \pmod n$$