The trinity doctrine of the 4th Century AD came about at the Council of Nicea in 325 AD. Those who gathered at the council have inferred from their interpretation of the bible that the Father, Son, and Holy Ghost are the same being but three separate persons. In the Church of Jesus Christ of Latter-day Saints, we have learned from the Prophet Joseph Smith that the Father, Son, and the Holy Ghost are definitely three separate and distinct persons but also separate and distinct beings. We know this from scripture since Jesus speaks of having to go to the Father after his resurrection.
John 16:10, 16
10 Of righteousness, because I go to my Father, and ye see me no more;
16 A little while, and ye shall not see me: and again, a little while, and ye shall see me, because I go to the Father.
Jesus proclaimed that he was not the same being as the father when he said that the Father was greater than he was.
John 14:28
28 Ye have heard how I said unto you, I go away, and come again unto you. If ye loved me, ye would rejoice, because I said, I go unto the Father: for my Father is greater than I.
Jesus and the father also have separate wills. Jesus, when in the Garden of Gethsemane, said, "Not my will, but thine be done". At that time Jesus wished that the cup of suffering could be removed from him but he humbled himself to the will of the Father even though he wished not to suffer as he did.
Luke 22:42
42 Saying, Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done.
Obviously the Father was in a different location since Jesus would have to go to his Father in Heaven. Obviously the Father and Jesus have separate wills and Jesus humbles himself to do the will of the Father. In other verses Jesus proclaims that the Father is greater than himself and that the Holy Ghost would be sent after Jesus leaves.
John 14:26,28
26 But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.
28 Ye have heard how I said unto you, I go away, and come again unto you. If ye loved me, ye would rejoice, because I said, I
go unto the Father: for my Father is greater than I.
when Jesus was on the cross he exclaimed:
Matthew 27:46
46 And about the ninth hour Jesus cried with a loud voice, saying, Eli, Eli, lama sabachthani? that is to say, My God, my God, why hast thou forsaken me?
If the Father and Son are the same being, how is it that Jesus felt forsaken?
It is true that the Father and Son are one. But this does not imply that they are the same being. They are one in purpose and truth. Jesus taught that we all can be one even as the Son is one with the Father:
John 17:20-23
20 Neither pray I for these alone, but for them also which shall believe on me through their word;
21 That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me.
22 And the glory which thou gavest me I have given them; that they may be one, even as we are one:
23 I in them, and thou in me, that they may be made perfect in one; and that the world may know that thou hast sent me, and hast loved them, as thou hast loved me.
Here Jesus teaches us that the oneness he has with the Father can be shared and that we all may be one even as Jesus and the Father are one. I don't think we will all become the same being. But we can share in the spirit, truth and light. The oneness Jesus has with the Father is the same oneness that we can have. It is not that they are the same being but that they share the same purpose, truth, glory, etc.
When Stephen was martyred he saw Jesus standing on the right hand of God.
Acts 7:55-56
55 But he, being full of the Holy Ghost, looked up steadfastly into heaven, and saw the glory of God, and Jesus standing on the right hand of God,
56 And said, Behold, I see the heavens opened, and the Son of man standing on the right hand of God.
One must ask how is it even possible to be 3 separate persons and yet the same being? I don't believe that there is evidence in scripture showing that God the Father, Jesus Christ the Son, and the Holy Ghost to be the same being. They do have a oneness but that oneness is not that they are the same being. They share the same purpose, glory, truth and light and as three separate and distinct persons, they are also three separate and distinct beings.