According to Trinitarians, if Jesus was saying things that clearly made Him out to be God, why didn't Caiaphas mention it at Jesus' trial?
Being instituted as high priest by Roman authorities, he needed a legal pretext to execute Jesus in the eyes of the Roman governor of the Roman province of Judaea: Pontius Pilate.
First of all, we have to remember that at the moment of Jesus’ trial there were two high priests in Jerusalem: Annas and Caiaphas and they needed a Roman legal reason to condemn Jesus to death, not a Jewish religious reason.
But they shouted, "Take him away! Take him away! Crucify him!" "Shall I crucify your king?" Pilate asked. "We have no king but Caesar," the chief priests answered. - John 19:15
Although Caiaphas was high priest that year, he was appointed to this position by the power and will of Rome, and not by the Jewish Sanhedrin. Thus it would be normal that some of the populace would still consider Annas to be the true high priest, since he was deposed by the Romans. We can see that the political circumstances surrounding Christ’s Passion were very complex.
Annas and Caiaphas are two high priests mentioned during Jesus’ public ministry (Luke 3:2). In that period of history, high priests were installed and removed by Roman rulers. While it is not recorded in the Bible, the tradition is that the Romans had deposed Annas and made Caiaphas the high priest. So, officially, Caiaphas, the son-in-law of Annas, was high priest during Jesus’ ministry, but Annas, the former high priest, still held significant sway and was still called a high priest (John 18:13).
When Jesus was arrested in Gethsemane, He was brought before Annas to be questioned (John 18:13; 19–23). Annas sent Him to Caiaphas (John 18:24). The Sanhedrin was also involved in this questioning (Matthew 26:57). Jesus was led away from Caiaphas’ house to stand before Pilate (John 18:28), who then sent Jesus to Herod (Luke 23:6–7), who returned Him to Pilate (Luke 23:11). Pilate eventually condemned Jesus to death by crucifixion, after declaring Him innocent three times (John 18:38; 19:4, 6).
Jesus’ trial before Annas and Caiaphas was marked by false testimony and conflicting reports of what Jesus had done and said (Mark 14:56). Through it all, “Jesus remained silent and gave no answer” (verse 61). Caiaphas began to despair of finding enough evidence to put Jesus to death, but then he asked Him directly, “Are you the Messiah, the Son of the Blessed One?” (verse 61). Jesus answered, “I am. . . And you will see the Son of Man sitting at the right hand of the Mighty One and coming on the clouds of heaven” (verse 62). At this, Caiaphas tore his clothes, decreed Jesus to be a blasphemer, and turned Him over to a mob who beat Him (verses 63–65). - What is the account of Annas and Caiaphas?
We must recall that the office of high priest, first conferred on Aaron by his brother Moses, was normally hereditary and for life.
Annas had been high priest from A.D. 6 to 15. The Romans had removed him from office yet he still wielded considerable power behind the scenes. Five of his sons succeeded him as high priest. He was the father-in-law of Caiaphas - the high priest who was in office at the time of Jesus ministry.
In the high priesthood of Annas and Caiaphas, the word of God came to John, the son of Zacharias, in the wilderness (Luke 3:2).
Annas is still called "high priest" even though he was not serving in that capacity at the time.
Jesus Was Brought To Annas First
His power was obvious. When Jesus was arrested He was brought to Annas first rather than Caiaphas.
And led him to Annas first; for he was father-in-law of Caiaphas, who was high priest that year. Now Caiaphas was the one who had advised the Jews that it was expedient for one man to die on behalf of the people (John 18:13,14).
The examination before Annas was unproductive.
The high priest then questioned Jesus about his disciples, and about His teaching. Jesus answered him, I have spoken openly to the world; I always taught in synagogues and in the temple, where all the Jews come together; and I spoke nothing in secret. Why do you question me? Question those who have heard what I spoke to them; they know what I said. When he had said this, one of the officers standing nearby struck Jesus, saying, "Is that the way you answer the high priest?" Jesus answered him, "If I have spoken wrongly, testify of the wrong; but if rightly, why do you strike me?" (John 18:19-23).
Annas then sent Jesus away to Caiaphas the current high priest.
So Annas sent him bound to Caiaphas the high priest (John 18:24).
Caiaphas then conducted the next phase of Jesus' trial.
Caiaphas was the ruling High Priest at the time of Jesus ministry (A.D. 18-36). He was the son-in-law of Annas.
Caiaphas predicted the necessity of the death of Jesus.
But one of them, Caiaphas, who was high priest that year, said to them, "You know nothing at all, nor do you take into account that it is expedient for you that one man die for the people, and that the whole nation not perish." Now he did not say this on his own initiative, but being high priest that year, he prophesied that Jesus was going to die for the nation, and not for the nation only, but in order that he might also gather together into one the children of God who are scattered abroad. So from that day on they planned together to kill him (John 11:49-53).
He is the one who plotted to kill Jesus.
Then the chief priests and the elders of the people were gathered together in the court of the high priest, named Caiaphas; and they plotted together to seize Jesus by stealth and kill him. But they were saying, "Not during the festival, otherwise a riot might occur among the people" (Matthew 26:3,4)
Caiaphas is the one whom charged Jesus with blasphemy.
But Jesus kept silent. And the high priest said to him, "I adjure you by the living God, that you tell us whether you are the Christ, the Son of God." Jesus ^said to him, "You have said it yourself; nevertheless I tell you, hereafter you will see the Son of Man sitting at the right hand of power, and coming on the clouds of heaven." Then the high priest tore his robes and said, "He has blasphemed! What further need do we have of witnesses? Behold, you have now heard the blasphemy; what do you think?" They answered, "He deserves death!" (Matthew 26:63-66).
Caiaphas sent Jesus to Pilate to have the death sentence carried out.
Then the Jews led Jesus from Caiaphas to the palace of the Roman governor. By now it was early morning, and to avoid ceremonial uncleanness the Jews did not enter the palace; they wanted to be able to eat the Passover. So Pilate came out to them and asked, "What charges are you bringing against this man?" "If he were not a criminal," they replied, "we would not have handed him over to you." Pilate said, "Take him yourselves and judge him by your own law." "But we have no right to execute anyone," the Jews objected. This happened so that the words Jesus had spoken indicating the kind of death he was going to die would be fulfilled (John 18:28-32).
Whatever the motivations were in regarding Caiphas’ intentions during the trial of Jesus, they were certainly influenced by the Sanhedrin and the religious, political situations of that moment in time. In an nutshell, it was complicated.