To get someone to trust you again, you should first apologize and show that you mean it. You can also try and use your actions to convince the person by not repeating what you did and by limiting your time away from them.
Lamentations 3:26-30 New International Version (NIV)
26 it is good to wait quietly for the salvation of the Lord. 27 It is good for a man to bear the yoke while he is young.
28 Let him sit alone in silence, for the Lord has laid it on him. 29 Let him bury his face in the dust— there may yet be hope. 30 Let him offer his cheek to one who would strike him, and let him be filled with disgrace. Cross references:Lamentations 3:26 : S Isa 7:4Lamentations 3:26 : Ps 37:7; 40:1Lamentations 3:28 : Jer 15:17; La 2:10Lamentations 3:29 : S Job 2:8Lamentations 3:29 : S Jer 31:17Lamentations 3:30 : S Job 16:10; S Isa 50:6Lamentations 3:30 : Mic 5:1
In other words, make yourself low and humble and wait. Be well behaved, slowly she will regain trust on you. Gby
Dont try to. Be trustworthy and they will eventually see it or hold a grudge. If you are talking about your parents... Never. They allow you rope to hang yourself with. They never trust you even if they say they do because they remember their childhood.