The phrase 'waste your breath' means using words with no result; it's not about actual breathing. I tried to change her mind, but I was just wasting my breath.
Pro Tip 2/3
Stubborn Audience
Use 'waste your breath' when someone won't listen or change their mind, despite your efforts. He'll never agree to that plan, so you're wasting your breath.
Pro Tip 3/3
Futile Situations
The phrase suits situations where effort has no chance of success. Arguing about it now is wasting your breath; the decision has been made.