No, you can't, because both superiority dice would apply to the same damage roll (for the same attack).
5e official rules designer Jeremy Crawford says the following (as quoted in Xanthir's answer):
Twitter user: Hi guys can a fighter use 2 maneuvers - Riposte on enemy's attack & Pushing Attack on his attack? i.e 1/attack
Crawford: The intent is no more than 1 maneuver associated with any attack. Riposte is in a gray area, but I would say no.
Rules on maneuvers per attack, from the Battle Master's Combat Superiority feature description (PHB, p. 73):
You learn three maneuvers of your choice [...] You can use only one maneuver per attack.
Feinting Attack says:
You can expend one superiority die and use a bonus action on your turn to feint [...] You have advantage on your next attack roll against that creature this turn. If that attack hits, add the superiority die to the attack's damage roll.
Trip Attack says:
When you hit a creature with a weapon attack, you can expend one superiority die to attempt to knock the target down. You add the superiority die to the attack's damage roll [...]
The use of your bonus action for feinting is an additional expenditure of resources to apply the advantage condition and superiority die to your next attack. The maneuver is still applied to the "next" attack and is not a separate event.
The Trip Attack modifies the attack to add a saving throw to the attack after landing a hit. It adds an a saving-throw property (to trip) and a superiority die to the damage. This saving throw is not an additional or separate attack.
If you "sandwich" them, you are applying two maneuvers to the same attack at the cost of two superiority dice and a bonus action to modify a single attack, which is not allowed. The litmus test for this is that if you were to use both, you would roll two superiority dice for the same damage roll. Which is not allowed.
These modifiers are similar to a paladin's Divine Smite ability. He is not making an additional attack to apply smite damage; he is giving the attack smite properties for additional damage. This is not an additional attack or spell separate from the attack, but expending spell slot resources to add a modifier to the same attack.
A cheap way to get advantage on all attacks AND still be able to use combat maneuvers is to take a two-level dip into barbarian. A character with at least 3 Battle Master fighter levels and 2 barbarian levels is able to use Reckless Attack AND maneuvers to modify the same attack. While raging, you also get a modest damage bonus.
On Wasted Superiority Dice
You can only expend a superiority die for trip AFTER the attack hits and before you roll damage. If you miss with your initial attack, you are not able to expend a superiority dice to add the tripping property to the attack.
So, even if you were able to use feignt and trip attack on the same attack roll (which you cannot), you would never waste the trip manuver as if you missed, you could not activate trip.