I apologize for things out of my control fairly frequently as a way of displaying (genuine) empathy for people in bad situations. Not so much for superficial things like everyday rain but say if someone is looking forward to an interview and doesn't get the job, I'll say something like "Sorry you didn't get the job".

I don't do this to build trust, but I guess it makes some sort of sense that if someone displays a consistent level of empathy towards the problems of others it would make them more trustworthy in general compared to someone who doesn't.