A young Severus Snape asked Voldemort not to kill Lily Potter. He knew Voldemort wanted to kill Lily's baby son, but asked that Lily be spared because Snape was in love with her.
“What request could a Death Eater make of me?”
“The — the prophecy . . . the prediction . . . Trelawney . . .”
“Ah, yes,” said Dumbledore. “How much did you relay to Lord Voldemort?”
“Everything — everything I heard!” said Snape. “That is why — it is for that reason — he thinks it means Lily Evans!”
“The prophecy did not refer to a woman,” said Dumbledore. “It spoke of a boy born at the end of July —”
“You know what I mean! He thinks it means her son, he is going to hunt her down — kill them all —”
“If she means so much to you,” said Dumbledore, “surely Lord Voldemort will spare her? Could you not ask for mercy for the mother, in exchange for the son?”
“I have — I have asked him —”
“You disgust me,” said Dumbledore, and Harry had never heard so much contempt in his voice. Snape seemed to shrink a little. “You do not care, then, about the deaths of her husband and child? They can die, as long as you have what you want?”
Why would he think Voldemort would keep that promise?
He likely knows Voldemort is not the type to keep any promise when it suits him to break his promises.
He likely knows Voldemort would kill full blooded wizards and even those closest to him.
He likely knows Voldemort would use Deatheaters, their house elves, and their properties as pawns.
He likely knows Voldemort might use knowledge of Snape's desire for Lily as a weapon against him. (Assuming Lily had lived that night.)
If he knows all this, then he knows promises mean nothing.