This question might have an obvious answer, but I need clarification: does a spell casting PrC which carries on one of the base class spell progressions, such as Arcane Trickster or Rage Prophet, gain the bonus spells that would be learned at the respective class levels, or does it only allow for the core spell progression of the spell casting class?
For example, does a sorcerer/rogue who takes Arcane Trickster still learn their bloodline bonus spells when leveling as Arcane Trickster? I would think no, but my players might argue with me on this one.
Nope. The bonus spells are a separate class feature (Bloodline), which does not improve with levels in other classes. You get spells per day and (for sorcerers and such) the usual spells known because they're specifically called out. You do not get bonus spells known from other features; you have instead opted to advance in some other way instead of developing your bloodline abilities.
Cosign what blahpers said.
You can compare the Dragon Disciple, which specifically continues Bloodline progression, to other casting PrCs.
And FAQ
There was a big thread where people argued in favour of prcs advancing bloodline spells. However like the two posters above me I disagree with this. A good sign something is not advanced by a prc is when its .e tionex repeatedly I the class table with its own entry. This is the case for both bloodline spells and mystery spells.