Can repeat cost be reduced with applied researcher

Yes — while Applied Researchers is Empowered, its discount reduces a Repeat cost, but only the one you actually choose to pay.

Repeat is an optional additional cost added to the base cost when you declare it (Rule 820.1), and discounts are applied to that combined total once (Rule 356.4.f — same treatment as Eager Apprentice, FAQ #629). So the Empowered discount — spells cost 1 Energy less (minimum 1) and 1 any-domain Power less — shaves the Repeat cost together with the base cost, not twice. If you don't declare Repeat, there's nothing extra to discount; and if the spell's Energy cost is already at the 1 floor, the Energy part of the discount won't help, though a Power-denominated Repeat (e.g. Repeat [A] or [C]) would still get the [A] Power discount.