Improved Trip uses some strange language:
IMPROVED TRIP [GENERAL]
Prerequisites: Int 13, Combat Expertise.
Benefit: You do not provoke an attack of opportunity when you attempt to trip an opponent while you are unarmed. You also
gain a +4 bonus on your Strength check to trip your opponent.
If you trip an opponent in melee combat, you immediately get a melee attack against that opponent as if you hadn’t used your
attack for the trip attempt.
Now, if you had true strike going, and tripped somebody, the true strike applies against the initial touch attack. Not the strength check. Ok. Got it.
What if you then MAKE the strength check, and the guy is tripped, and you have imp. trip, triggering the free attack?
You "gain a melee attack against that opponent as if you hadn’t used your
attack for the trip attempt."
Does this mean you get the True Strike +20 once again, plus 4 for the opponent being prone?