$\begin{align*}\frac{{n \choose 2}{n-2 \choose 2}{n-4 \choose 2}\cdots {2 \choose 2}}{(\frac{n}{2})!} &= \frac{\frac{n!}{2^{n/2}}}{(\frac{n}2)!} = \frac{\frac{(n-1)!! \cdot n!!}{2^{n/2}}}{(\frac{n}2)!} \\
&= \frac{(n-1)!! \cdot (2^{n/2} \cdot (\frac{n}2)!)}{2^{n/2} \cdot (\frac{n}2)!} = (n-1)!!\end{align*}$