I've found a simpler proof for some identity/theorem, hypothetically speaking, of course ;) How do I know if it hasn't been done before? For important results it's fairly easy to find.
By the way, I think I'm the first one to use the "alternative-proof" tag. I feel original today.. lol
I do not know what results you might have in mind, so I will try to be as general as possible.
No matter how important result is, it is a good idea to check it to make sure you have done everything correctly. The excitement of proving something can blind us to our mistakes. If this checks out it really depends on the result how you take it from there.
Say you have proved a theorem or lemma from a paper in a nice way. There is a high chance it has already been done. Was the idea intuitive? Did the proof write itself? If yes, then it has probably been done.
If you have proved or improved on a big result, you should probably check over your work again to make sure. If you do not want to speak to another person out of fear that they might scoop you, one thing you can do is check out survey papers on the topic at hand. Often, surveys will mention when a result has been proven in new interesting ways.
The other thing you can do is contact experts in the subjects. You do not have to "give away" the proof to talk about it.
