Depends. I once fell a dozen or two dozen times in one wet and muddy mountain/trail race running in road shoes. That sucked.
If it's dry, road shoes are fine. Some trail shoes are not good in some trail conditions too... there have been some with poorly chosen outsole compounds that are slippery when wet (read reviews of the shoes you might be interested in, so you won't be surprised).
Some road shoes have traction as good as some trail shoes. The road Nikes that I used as a kid had waffle soles that look like current trail shoes. Other road shoes with low profile, blown rubber outsoles have terrible traction in mud. Many trail shoes are totally fine on asphalt.
Shoes are shoes, and as long as they aren't super specialized, I wouldn't consider it a waste of money to get a trail shoe as an additional trainer. It will get used up just the same.