It's a very hard problem, in general. Many mathematical results have been re-proved over time. The Internet has helped somewhat by making some resources (e.g., the OEIS and the arXiv) available and making communication easier (e.g., by a search engine or this forum).

Best is to learn the relevant field enough to know the terminology, then searching for the result, and finally by asking an appropriate resource (forum, mailing list, newsgroup, etc.).