How can you earn your mother's trust back?

That would depend on how it was lost in the first place. Truthfully, regaining lost trust is unlikely in most situations, at least to the level at which it previously existed. This is a common aspect of human nature, consider the adage: One burned, twice shy. Your best hope is to not try, rather not to try falsely. Just be you and behave in a respectful, and honorable way. It may happen, but do not become discouraged if it takes a very long time, or not to the desired level.