Someone spread rumors verbally against someone else and the only way to prove that the first person is a liar is to post his recorded phone call conversation with the second one.
Colorado is a a one recorder state and so you can post it online. I just would not make ANY editorial comments about the recording whatsoever.
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Contracts / Agreements Lawyer
A person who is party to a phone call can generally record it. If you are not a party to a phone call, you generally cannot. However, it's probably unwise to post such things. There are probably better ways to resolve the issue.