The FRCS and PGtF aren't considered core books. 
 
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'Players Guide to Faerun' 
  
  
  
Faerun is a world on its own, just like all other ones.  
Because the so-called 'Player's handbook' doesn't have most of the reaces as playable, I will not consider it as the Bilbe of RP-ing, and won't use the Greyhawk gods for all my clerics. That book was written clearly  by people who use mostly the Greyhawk world for their story. 
 
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Um, no.  Only the PHB, DMG, and MM (I) are the core books.  Everything else is considered by WotC to be optional. 
 
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But see, they are for Greyhawk, even if not named like that. Even the gods listed in them are from Greyhawk. 
 
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And?  Your point?  They are the core rulebooks as defined and stated by Wizards of the Coast.  Josh's point is entirely correct.