Regaining someone's trust after you've lied isn't easy. Can it be done? Yes, but it depends on a lot of factors.
First off, if you demonstrate a big pattern of lying, I hope the other person dumps you because they shouldn’t have to live out the rest of their lives unable to trust the person they are supposed to lean on most in the universe.
Second, a lie is a lie.
I don't care what you lied about. A lie breaks trust, and it's a big deal, no matter how inconsequential you’d like to make it seem.
The first thing you have to do is take full responsibility
. No excuses! Nobody is going to know if you have true remorse other than you, but you have to take responsibility and promise that you’re willing to work hard to earn back their trust. Don't pile lies on top of lies trying to fix the original lie.
Also, tell the truth about why you lied. For example, say, "I lied because I didn't want to look bad," or, "I lied because I didn't want to get caught." People are more willing to believe you when you're willing to bare your soul.
For a while, you are going to be the bad guy or gal. Deal with it – you earned it. Don’t get on a high horse and say, "I apologized, can't you let this go?" The other person has to rebuild their trust and belief in you, and they're not going to do it just because you say they should. They already know you're a liar and that what you say means nothing. They need to watch you over time. And by time, I don’t mean minutes. It might take weeks or even years.
Over time, you need to be reliable and an open book. You must respect the needs of all the people you’ve hurt. If they need to whine, complain, and give you crap for a while, you're going to have to tolerate it.
Finally, stick to your promises. Don't make a promise that you're not 100 percent sure you can follow through on.
Any promise you make that you don’t live up to loses you credibility. The person you've hurt doesn’t need you to yap your way out of it – they need you to live your way out of it.