If I am playing Lux legend and I have an applied researcher in the back line making my falling comet cost four instead of five do I still get to draw one off of lux legend?

Yes, you still draw 1.

Lux's Legend ability checks the printed cost of the spell (the number in the top left corner), not the reduced cost you actually pay. Falling Comet has a printed Energy cost of 5, so it meets Lux's threshold regardless of Applied Researcher's discount. Cost reduction effects like Applied Researcher lower what you pay but don't change the card's printed cost for trigger purposes (Rule 131.4).