Battle Master fighter's Trip Attack when the target is already dead from the regular hit?

Question

The Battle Master fighter has the Trip Attack maneuver (PHB, p. 74):

When you hit a creature with a weapon attack, you can expend one superiority die to attempt to knock the target down. You add the superiority die to the attack’s damage roll, and if the target is Large or smaller, it must make a Strength saving throw. On a failed save, you knock the target prone.

By R.A.W., you wait to see if the attack hits before rolling the superiority die for Trip Attack.

Is there a rule which demands that the DM resolve the damage and determine whether the target is dead, before a player ends up wasting their superiority dice on a pointless maneuver?

Answer 1

Not dead yet

As you've quoted, the Maneuver is added on the hit and then you add the superiority die to the damage and resolve the save.

Damage hasn't yet been rolled, so you don't know if the regular hit killed a creature.

This does mean that you may expend the resource with no additional gain.

Answer 2

You shouldn't waste the die

How to be a mean DM 101:

1. Wait for the Fighter to land a killing blow
2. Ask the player "will you spend any superiority dice"
3. When the player spends any dice, say "oh, well, you've actually killed it with this strike, so you've just wasted the die, congratulations"

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In 5e, hitting and dealing damage is a single atomic operation. It is perfectly fine to make both rolls (attack roll and damage roll) at the same time — a legit way to speed up combat.

if the target is Large or smaller, it must make a Strength saving throw

This also assumes the target has to be alive in the moment of the maneuver. The maneuver itself seems pointless (kill somebody and then trip the corpse) fiction-wise as well. However, a DM can say otherwise.

Is there a rule which demands that the DM resolve the damage and determine if the target is dead

There is a rule (kind of) from the AL DM guide:

Always follow this golden rule when you DM for a group: Make decisions and adjudications that enhance the fun of the adventure when possible.

Making the player to waste resources due to unnecessary lack of information wouldn't be fun, I guess, so I strongly recommend against it. Your DM might say otherwise though, so perhaps you should speak with him/her in advance.