A fighter receives a superiority die for choosing Superior Technique fighting style, then reaches level 3 and chooses Martial Archetype; Battle Master.
Can the superiority die from superior technique be used to perform the newly learned maneuvers?
Relevant Rules info:
SUPERIOR TECHNIQUE You learn one maneuver of your choice from among those available to the Battle Master archetype… You gain one superiority die, which is a d6 (this die is added to any superiority dice you have from another source). This die is used to fuel your maneuvers. (Tashas p.41)
And
COMBAT SUPERIORITY When you choose this archetype at 3rd level, you learn maneuvers that are fueled by special dice called superiority dice…
Superiority Dice. You have four superiority dice, which are d8s. A superiority die is expended when you use it. You regain all of your expended superiority dice when you finish a short or long rest. (PHB p.73)