It is a theological and apologetic error to try to “prove” the Trinity.
We can readily offer our good reasons for why we believe it, with all the scriptures and teachings of the Church, and we may be certain of its truth by our own faith and reason, but it is a divine mystery that cannot be proved by mere intellectual reason alone, and anyone who asserts they can prove it makes a terrible apologetic error and will get shot down by his opponents.
Stick to the teachings of the Church and her interpretations of scripture to explain and defend why we believe in the Holy Trinity, but we should never assert that we can prove it by human reasoning alone because it completely defies any vestiges of such proofs and makes its proponents appear foolish.
Mark Brumley calls this the first deadly sin of apologetics. Namely
"Apologetical Gluttony"
"…Biting Off More than You Can Chew. (See discussion:
Mark Brumley’s “How NOT to Share Your Faith”)
