Identify the italicized phrase as a participial, gerund, infinitive, or appositive phrase.

Mervin, the circus animal trainer directed the elephant by tapping her trunk.

You didn't italize or mark the phrase, but I see one good candidate:

"The circus animal trainer" is in a way another name given to Mervin, a kind of "renaming" him: this is called an appositive phrase, so if this was the phrase appositive phrase is the answer! (also, I don't see the other phrases here).