Determinism is the view that whatever happens is determined by antecedent conditions under the aegis of the laws of nature. Equivalently, past facts, together with the laws of nature, entail all future facts. It follows that facts before one's birth, via the laws of nature, necessitate what one does now. The necessitation here is causal, not logical. Could a determinist have a use for 'could have done otherwise'? Yes, if he gives the phrase a weak or conditional interpretation. No, if he gives it a strong or unconditional interpretation.
WEAK READING. Agent A could have done otherwise than action X =df A would have done other than X had A had a sufficiently strong desire to do other than X (or had a sufficiently strong desire together with a different set of background beliefs, etc.)
Example. A man insults me and I insult him back. Could I have "turned the other cheek" and done otherwise? Yes, under conditions like the following. Had I been a better man, I would have let the insult pass unanswered. If I had not perceived the insult, I would not have answered it. If I had had a desire to impress a bystander with how forebearing I am, I would have remained silent. And so on.
STRONG READING. Agent A could have done other than X =df A could have done other than X even if every factor prior to X had been the same.
I will use 'could have done otherwise' only in the STRONG sense. This will allow me to define libertarian freedom (L-freedom) in terms of 'could have done otherwise': An agent A is L-free in respect of action X =df (i) A performs X; (ii) A could have done otherwise. It is clear that L-freedom is incompatible with determinism. For if I am L-free in respect of just one action, then it is not the case that whatever happens is causally necessitated by antecedent conditions via the laws of nature.