The most convincing evidence for the Trinity lies in the truth that God is love (1 John 4:8 & 16). To love is to be in a relationship with others. Since God is eternally loving, it makes sense that even before the creation of anything, there was someone to love. But how can that be since God is one? Well, what would you call three divine beings perfected in love? When you consider a loving marriage that has existed for fifty or sixty years, have not the two essentially become one flesh? How do you identify the one without the other? How does one exist without the other? Just like the shapes of individual pieces of a jigsaw puzzle become lost in a completed image, the lines of individuality fade into the background when the love of two or more beings is perfected (1 Corinthians 13:8-12).
Since the Father, the Son and the Holy Spirit are unified through the eternal and perfect application of divine love, how do you separate them? Are they not essentially one?
The doctrine of the Tri-unity works because God is love. And because God is love, we also can participate in that union by becoming a Christian and reciprocating that love through obedience (John 14:20-21).