While recently developed Angluin-style learning algorithms for omega-automata have much in common with her classic DFA learning algorithm, there is a huge difference in the cost of the equivalence queries about the target automata. For omega-regular languages, the target is to learn nondeterministic Buchi automata (NBAs) through the vehicle of Families of DFAs (FDFAs). While the cost of equivalence queries is usually idealised as constant in learning, it makes a practical difference that the language equivalence checking about the learned NBAs is computationally hard. We develop efficient techniques for the cases, where we learn deterministic Buchi automata (DBAs) or deterministic co-Buchi automata (DCAs). This is based on the observation that some classes of FDFAs can be used to learn DBAs for DBA recognisable languages, rather than having to resort to nondeterministic ones. We believe that the restriction to DBAs and DCAs in equivalence queries also makes our algorithm more appealing to realistic applications, as the operations are cheap---NL---for DBAs and DCAs.