What do you do if someone thinks you're lying when you're not lying?

Give thanks and praise and trust to God.

1 Peter 2

19 For this is thankworthy, if a man for conscience toward God endure grief, suffering wrongfully.

20 For what glory is it, if, when ye be buffeted for your faults, ye shall take it patiently? but if, when ye do well, and suffer for it, ye take it patiently, this is acceptable with God.

21 For even hereunto were ye called: because Christ also suffered for us, leaving us an example, that ye should follow his steps:

22 Who did no sin, neither was guile found in his mouth:

23 Who, when he was reviled, reviled not again; when he suffered, he threatened not; but committed himself to him that judgeth righteously:

24 Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed.

25 For ye were as sheep going astray; but are now returned unto the Shepherd and Bishop of your souls.

1 Peter 4

12 Beloved, think it not strange concerning the fiery trial which is to try you, as though some strange thing happened unto you:

13 But rejoice, inasmuch as ye are partakers of Christ's sufferings; that, when his glory shall be revealed, ye may be glad also with exceeding joy.

14 If ye be reproached for the name of Christ, happy are ye; for the spirit of glory and of God resteth upon you: on their part he is evil spoken of, but on your part he is glorified.

15 But let none of you suffer as a murderer, or as a thief, or as an evildoer, or as a busybody in other men's matters.

16 Yet if any man suffer as a Christian, let him not be ashamed; but let him glorify God on this behalf.

17 For the time is come that judgment must begin at the house of God: and if it first begin at us, what shall the end be of them that obey not the gospel of God?

18 And if the righteous scarcely be saved, where shall the ungodly and the sinner appear?

19 Wherefore let them that suffer according to the will of God commit the keeping of their souls to him in well doing, as unto a faithful Creator.

you live a good life so that they will see your personality and trust in you that way

Don't do anything. If they don't believe you, then they don't trust you, why associate with folks that don't trust you?

Not lying

Start looking for proof to use!

At that point you can only pray for them, pray that their spiritual eyes of understanding be enlighten...

Live your life in accordance with what you know to be true.

This happens to me from time to time. I have learned from my dad to make my statement ONCE and shut up. You can't control HOW OTHER people are.

One day they will need someone to believe them... that's when you get to kick their A55. Some people need to experience the effects of their own punishments, before they learn fairness or right.

That would depend on whether they're being reasonable in not trusting you. If they have a good reason, it's probably best to be completely open and honest without pushing them to trust you. If they don't really have a good reason, it's probably either a trust issue on their part or just them being unreasonable, and it's probably better to ask why they don't trust you.