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Trip (Ex)
An eidolon becomes adept at knocking foes to the ground with its bite, granting it a trip attack. Whenever the eidolon makes a successful bite attack of the selected type, it can attempt a free combat maneuver check. If successful, the target is knocked prone. If the check fails, the eidolon is not tripped in return. This ability only works on creatures of a size equal to or smaller than the eidolon. The eidolon must possess the bite evolution to select this evolution. Source: Advanced Player's Guide
So, I have two questions.
First, if the damage from the bite is less than the target's damage reduction, I would say that the bite attack is not "successful", so I cannot take the free trip.Second, if there is a miss chance, does the miss chance apply separately to the bite and the trip? It seems to me that if the bite hits, then the miss chance should not apply to the trip.
If an attack does not deliver any damage due to DR, any secondary effects of the attack do not occur either. That's in the rules somewhere, and I'd say it applies here.
The trip has it's own roll, so it gets its own miss change. I must admit it doesn't make much sense, but those are the rules as I know them.
if the damage from the bite is less than the target's damage reduction, I would say that the bite attack is not "successful", so I cannot take the free trip.
Damage Reduction: "Whenever damage reduction completely negates the damage from an attack, it also negates most special effects that accompany the attack, such as injury poison, a monk's stunning, and injury-based disease. Damage reduction does not negate touch attacks, energy damage dealt along with an attack, or energy drains. Nor does it affect poisons or diseases delivered by inhalation, ingestion, or contact."
I think the bite attack was successful, in that it hit the creature. The trip is triggered by the attack landing, not by the damage it deals.
Second, if there is a miss chance, does the miss chance apply separately to the bite and the trip? It seems to me that if the bite hits, then the miss chance should not apply to the trip.
The trip is triggered by the attack, but it's still a combat maneuver check, which means it's an attack with a separate attack roll. As such, I think anything that applies to a single attack roll (True Strike, or any one-shot bonus, or invisibility) would not apply to the second attack roll to make the combat maneuver. So I think the miss chance would apply to both attacks.