What is the female equivalent of Duke?

Answer 1

A female Duke can be a Duke or Duchess. The wife of a Duke is a Duchess

Duke is unique amongst the British titles in that if a woman holds the title in her own right the title may be the same as the male (unlike, say, Baron, where a woman given the same title is always a Baroness).

So while Kate Middleton is the Duchess of Cambridge (because she is married to the Duke - William), the Queen is the Duke of Lancaster (because she holds the title, not Philip). The Queen is also a Duchess as a result of being married to a Duke.

Answer 2

Duchess.