Let's say I was writing a proof that required a lot of leg work or "rough work" in order to arrive at the conclusion. I want to show the reader (say a professor) that rough work because it's important. The actual proof after the work is trivial but long. Is it legitimate for me to (after the work) say something like, "Choose $x = \dots $, and the result will follow."?
I would say it depends on how 'trivial' it is. Triviality can be subjective. You can say it, but you could also include just the general idea of the remaining steps. For instance, "choose x=... and the result will follow by simple algebraic manipulations". Not always necessary, of course, but it puts you in a 'safer' position, so to speak.
In my view (and this is not shared by everyone!) the correct way to do this is to put into an appendix as much detail as you feel would let someone actually verify your assertion easily. Yes, this might make your paper/answer/whatever fifty times longer; but bytes are cheap.