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)