Number of spanning trees in undirected simple graph
You cannot show this since it isn't true. I encourage you to try out some actual numbers (e.g. $n=2$) and see for yourself that the numbers don't match. The problem is that what you should be subtracting is not the number of cycles of length $n-1$, but rather the number of collections of $n-1$ edges which contains at least one cycle. Such a collection need not be a cycle of length $n-1$.

In addition, the formula for the number of cycles of length $n-1$ is wrong. The number is $n(n-2)!$ rather than your $n(n-1)!$, and even this formula is only valid for $n \geq 4$ (for smaller $n$ there are no such cycles). However, even after replacing $n(n-1)!$ by $n(n-2)!$, the numbers don't match.