Well first things first, whether or not you're looking for your spouse's trust, you're supposed to share all relevant information with them. You're a team and if one person is misinformed then the team is misinformed. In my experience transparency is the best thing for a relationship to succeed and in most cases if it's not there then the relationship will fail.
Secondly, trust in a relationship is something that you shouldn't have to go looking for, but if you find yourself in a position where you need to gain your spouse's trust then not withholding any information from them is probably the best way to do it. I mean, trust is earned and if being open is what's required then that's what you should do.
The one thing that tears relationships apart the most are secrets, I mean there are certain things that your spouse is better of not knowing, but as long as you decide to be with someone then their business is your business and yours is theirs so any information that you have should reach their ears as well.
I don't see any reason why I'd hide anything from my spouse, experience and even movies have proven that it's a terrible idea. Besides, whatever you withhold from them may eventually still get to their ears and they'll be left wondering why you didn't tell them about it.
I hope this helps.