analyse
2sinCcosA=sinA(sinC-2cosC)
∴2(sinCcosA+sinAcosC)=sinAsinC
∴2/tanA+2/tanC= 1
Use sinAsinC on both sides at the same time.
∴ 1/2= 1/tana+ 1/tanc
≥2 √( 1/tanA 1/tanC)
=2/√(tanA tanC)
∴√(tanA tank) ≥4
∴tanA tanC≥ 16
Obtained when tanA=tanC=4.