How should I tell a professor the answer to something he doesn't know?

I think there is an easy way to phrase this to make it tactful, after all professors are often curious about learning new things too!

"Hello Professor X, I hope you've been well! Last year I asked you a question and we couldn't figure it out at the time but I've since come across an interesting answer and just wanted to pass along the info just in case you're curious. We were discussing Y ... "

As an alternative to Juan's answer (roughly in the same spirit), I suggest that you can phrase the email regarding the issue itself--since presumably the professor would also be interested in knowing the answer to the problem. Something like the following should suffice.

Hi Professor XY, recently I learnt about [...], which seemed really interesting to me because [...]. If you recall, this is similar to what we discussed a while ago regarding ABC, which is what prompted me to look into this further. I thought I would send this to you in case you happened to be interested in it. Have a nice day!

The point is to focus on the part which the professor would also be curious about/interested in, and to not dwell on the fact that they weren't aware of the answer beforehand.

The polite way to do this in academia is to pose it as a question. This way, you show your humility. You acknowledge that the solution or answer you found might be flawed or incomplete. Also, you open yourself up to collaboration.