Unless your brother buys the game himself or plays it on your account, he'll only be able to play the trial. Buying the game has given your account the license to play the full version. Since your brother hasn't bought it on his, he can't play anything but the trial.
It's not enough for people to just live. They have to live for something.
Funny, my son bought the game using our shared xbox, and I can play the full game. We do both have gold accounts though.
I hear voices.....and they don't like you....
If you try playing Minecraft with an offline account, you get a message reading that "this games has some features which require an Xbox LIVE enabled gamer profile" (i.e. online multiplayer and leaderboards) and then lets you choose to connect to XBL or continue playing offline. However, you can still play offline and not just the trial version if you select "continue playing offline". For some reason you do get that message a second time, though, so just select "continue playing offline" again. You can then create new worlds and play them, change your skin if you've downloaded any skin packs, etc. And I've just tried that myself with one of my offline accounts - works like a charm.
First Law for Programmers: A programmer must not harm the framerate or by omission of action allow the framerate to be harmed.
Anyone on the xbox console that you bought the game on can play it. Online or offline. if your brother is playing on the console you actually purchased minecraft on, he can play. If he has another console, no he will have to buy it. But heck, its $10. Just buy it.
Make sure he is trying to launch the full game, not the trial.
"Understanding women isn't about knowing what button to push, its knowing when not to push them"-Me