I am in a class where we learned this, but my teacher didnt make us prove these properties formally. What we did was show that if we have two NFAs and which recognize the languages and , respectively, then we can create a new machine which recognizes by creating a new start state with an transition to the start states of and . This way, the new machine will nondeterministically recognize or . Then, since we have constructed an NFA which recognizes the language , it must be regular, since NFAs are equivalent to regular expressions. I will go through my book to see how you formalize the argument. Sorry if this doesn't help.
edit: OK, so your next step should be to create a new start state and describe your new transition function with and keep and the same. Then you make . You should also make . Hope this is correct and helps.
edit 2: I just saw you are working deterministically, so if you haven't learned about NFAs then this is kind of a useless post.