A family member thinks I am lying about where I am going but I'm not, what should I do?

Communicating with them is very crucial. It would be helpful to ask their permission when going out. Moreover, updating them about what happened or recounting details to them will help.

Show them the proof of where you are going, only to reassure them, then ask them to please trust them in future, As you are responsible and proving responsibility to your parents and ALWAYS being honest with them. They may sometimes not agree with something but at least they know you are reliable in telling them the truth.

Try to talk it out with them calmly. Explain that you really are being honest and that them not believing you is hurtful. However, you can't make them believe you, and so the only thing you can truly do then is live with the fact that you know the truth

I think that sometimes it's normal to feel suspicious, even though it's not an okay thing to do. The smartest thing you can do is to prove to them that you aren't, for example, if you've gone to the mall with friends, you can send them pictures of exactly that with the location attached to it.

While telling you what to do can be harmful and may not always work in this situation, I can suggest putting one of those location trackers on your phone and showing your family member where you're going if you're comfortable (like with iPhones you can send your global position to prove where you are based on your phone's internal GPS).

If it is someone that does not have influence on your curfews or other things that could negatively afffect you, just ignore them. If it is someone like your parent, maybe send them pictures when you arrive.

You don't have to prove anything, you are innocent. If they want to think you are lying, leave them at it, its no skin off your back. Be proud of yourself, you are honest and true, the rest of the world can think what they want. This is also the best way to prove yourself too, if you try to argue and convince this person that you are not lying, it will seem like you are hiding something, but if you just carry on as usual not worrying if they think you are lying, you will appear far more truthful and they are more likely to believe you!

It sounds like their behavior is a bit controlling, but if you are okay with it, maybe find a way to verify your location when you go out so they will see you are telling the truth. If they continue to ask you after you do that a few times, then that is their problem completely at that point.

Discuss with them why they feel that way towards you and that it makes you feel upset. Try to come up with a solution together to overcome this issue.

The best way to let them know you're telling the truth is to provide proof; take pictures of yourself at this place, or get someone to go along with you so they can vouch for you.