They did, just not in the same way than European that used full plates.
The technical features that allowed the development of plate armor were issued from Arabo-Islamic world, so I doubt they would need to "reverse-engineer" it.
It's just it was adapted to their tactics, their terrain (even Reconquista fighter didn't used full plates) and used something more close to cuirasse than anything, or using more often part of armor than full set.
The westerneuropean tactic of heavy cavalry man was "guided missile". At the end, you have an human tank that doesn't even need a shield.
The middle-eastern tactic, however, was based on mobility and, as RGB said, full plates isn't what you search then.
Basically what RGB said. It's first about tactical differences, then geographical context.
There could be lots of reasons for that, from tactical to economic. Spanish soldiers in the New World used mail, scale and brigandine armour all the way into the mid-17th c. for example, even if domestically mass-produced plate was far more common.
Cause you don't fight in a southern Mediterranean mountainous terrains OR in sub-tropical mountainous/foresty terrain in full plate. Not unless you want to become a puddle.
In this regard, Spanish equipment was kind of a middle way between western Europe and Mediterranean warfare.
I would point as well there's no need of retro-engineering considering the technical advances that allowed plates in Europe came from Arabo-Islamic world where a metallurgic "industry" -fail of more fitting word- existed before European one.
In plates, in gunpowder artillery, etc.
I'll just disagreeing with RGB on this point
:
1. Against anything but guns, plated mail is plenty good enough.
You had fabrication of plate armors up to the XVIII where they effectively protected against guns. Their main issue wasn't their effectiveness, but their weight that reached 35 kg that was suicidal when move was more and more privileged.