(i) State Lagrange's Theorem, and prove that, if $p$ is an odd prime,

$$(p-1) ! \equiv-1 \quad \bmod p$$

(ii) Still assuming $p$ is an odd prime, prove that

$$3^{2} \cdot 5^{2} \cdots(p-2)^{2} \equiv(-1)^{\frac{p+1}{2}} \bmod p$$