For example: a multiclass Fighter 6 and Unchained Barbarian 6 makes a full-attack action against a target. In rage he has has +12 BAB, +6 STR, +2 attack bonus from rage and a +3 Greatsword. Would he attack with +23/+18/+13 with his Greatsword, +15 from Steel Headbutt and also +15 from Animal Fury?
A creature that takes a normal full attack action can freely mix melee attacks, ranged attacks, and combat maneuvers counting as attacks
No limitations are placed on what kind of attacks can be made during a full attack, the action specifying only attacks and not requiring, for example, all attacks to be melee attacks because the first attack was a melee attack. (I'm not sure how else such a restriction would be adjudicated—good thing it doesn't matter!) During a full attack combat maneuvers that count as attacks like disarm, sunder, and trip can also be mixed freely alongside those melee and ranged attacks, but combat maneuvers taking a standard action like grapple and overrun can't.
Hence any creature—zen archer monk or not—that takes a full-round action to make a full attack (and not a flurry of blows) can lead with the combat maneuver trip (and possibly provoke attacks of opportunity) then follow that with ranged attacks (and, if threatened, possibly provoke more attacks of opportunity). (Also, such a creature will suffer a −4 penalty on ranged attack rolls made against that prone target.)
The only real advantage a zen archer monk has in using such a tactic is that, due to the monk's extraordinary ability unarmed strike, such a monk really doesn't need to take a free action remove one hand from his bow to attempt the combat maneuver trip then take a free action to put his free hand back on the bow to make the ranged attacks. Whether other creatures attempting the same routine must go through this rigamarole is up to the GM.
Although the character's trying really hard, he still can't wield a too-big two-handed weapon in one hand
What stops the Medium titan mauler barbarian 3/titan fighter fighter 1 from wielding in one hand a Large two-handed weapon—despite his many special abilities allowing him to wield weapons of unusual sizes—is the restriction on the extraordinary ability jotungrip of the archetype titan mauler, which says
At 2nd level, a titan mauler may choose to wield a two-handed melee weapon in one hand with a –2 penalty on attack rolls while doing so. The weapon must be appropriately sized for her, and it is treated as one-handed when determining the effect of Power Attack, Strength bonus to damage, and the like.
Emphasis mine. The titan mauler archetype's extraordinary ability massive weapons allows the warrior to use 2-handed weapons that are inappropriately sized more easily, and so does the titan fighter archetype's extraordinary ability giant weapon wielder, but neither special ability changes how the special ability jotungrip functions, and that special ability functions only on appropriately sized weapons, and that's the one that would need to change for the character to use in one hand a two-handed too-big weapon.