I saw someone who did one on Gravemind so I will do one on Tartarus.
Think about it, Tartarus must have been annoyed with the Elites and for good reason. For hundreds of years the Brutes were always treated as a lesser species and were usually called stupid because of their attitude, and when they got the chance to break free of their unfair treatment they let loose on the Elites, wouldn’t you if you were treated badly by them for centuries.
Another thing, Thel 'Vadam was accused of heresy but he was then promoted and was ranked even higher than Tartarus and he had commited no treason but someone who had let one of the Sacred rings fall to the humans was allowed to be promoted.
When the brutes got a well deserved promotion from the Prophets since one of the High Prophets was lost because of the Elites, the Elites then left the Covenant. The Brutes never left the Covenant when times were hard but as soon as the Brutes were promoted they left.
Lastly, Tartarus was just listening to orders put by Truth. He may have not wanted to kill the Arbiter but he was told to and if he didn’t accept his orders he would have been killed himself. And he was told to activate Halo he did because he had to obey them. Many people would probably say that he should have listened to 343 Guilty Spark when he was told to but if you think about why should he? Tartarus was probably tired of all the pathetic excuses from the Elites and slowing down the Great Journey which he was taught to believe.