Using a switch on before a 2nd Router

> 3. cat 6 cable running into Net gear Router # 2(Net Gear Night Hawk AZ
> 1750) in my living room.

   "AZ 1750" ("AC1750"?) is a speed, not a model number.  Ever
helpful, Netgear product packaging emphasizes speeds like "ACxxxx", but
that's not the model identifier.

> [...] Apple TV won't pick up my itunes library and I believe it's
> because it's not on the same router. [...]

   Where is "my itunes library"?  How would it know where to find "my
itunes library"?

> f I put a switch on the cat 6 line before it went into router #2, then
> would everything on the switch be considered to be part of router #1?

   Not "part of router #1", but on the LAN of router #1.  I don't know
enough about how an Apple TV finds an iTunes library to say whether that
would matter.  I'd expect a device on the LAN of router #1 to be
visible to a device on the LAN of router #2 (but not the other way
around, unless a suitable static route were added on router #1).

> If I put a switch on the cat 6 line before it went into router #2,
> then would everything on the switch be considered to be part of router
> #1?

   Part of router #1's LAN, yes.  If you want the wireless capability of
router #2, then it might be simpler to configure router #2 as a wireless
access point.  That would make everything part of router #1's LAN
(without adding more hardware).  I'm not sure what that would do to the
more exotic features of router #2 (ReadySHARE, for example).

> Would something like NETGEAR GS105NA 5-Port Gigabit Ethernet Network
> Switch do the trick?

   Sure, but if you've see one cheap-Chinese gigabit Ethernet switch,
you've pretty much seen them all.

> Does using a switch slow down my Ethernet line?

   It would add a tiny bit of latency, probably unnoticable.

   Visit http://netgear.com/support , put in the model number of router
#2, and look for Documentation.  In the User Manual, look for "Wireless
Access Point".  Or, perhaps, in the web interface: ADVANCED > Advanced
Setup > Wireless AP