[Retsu Unohana]

Does bromination of a cyclic compound occur the same way as a simple aliphatic compound?

For example, if you brominate 1-tertbutyl-cyclohexane, do you get 1-bromo-1-tertbutyl-cyclohexane? Or do you get the bromine elsewhere on the ring?

[azmanam]

depends on what the conditions are.  HBr?  Br2?  NBS?  light?  heat?

[Retsu Unohana]

I think bromination uses Br2 every time. In that case, wouldn't there just be a single major product?

[azmanam]

Not necessarily, but I misread the question at first. I agree with you that Br2 will be the conditions here. In that case, I agree with your answer in op.  Why do you think the Br will add there?

[Retsu Unohana]

Well tertiary carbocations are the most stable, so I believe the bromine will add there. I'm just thinking whether or steric hindrance by the t-butyl will affect the reaction.

[azmanam]

tertiary carbocations are the most stable

True, but there will not be a carbocation intermediate in this mechanism.  Start here:
http://www.cem.msu.edu/~reusch/VirtTxtJml/funcrx1.htm#halogen

I'm just thinking whether or steric hindrance by the t-butyl will affect the reaction.

eh, maybe.  I think the major product will be the one you describe.

[Retsu Unohana]

^ Sorry, I meant tertiary radical.

But thanks for confirming my answer.

[bacrcr]

different reaction conditions may results different products ，especially for this sensitive
reaction