When Ziva first shows up, Director of NCIS Jennifer Shepard (Lauren Holly) vouches for her trustworthiness, telling Gibbs about Ziva saving her life earlier on a mission in Cairo. But of course, Gibbs has to decide for himself whether or not Ziva is worth his trust, and their relationship doesn’t really start off well.
After Kate’s death, Gibbs has an intense vendetta against Ari. Then here comes Ziva, who attempts to help Ari escape with falsified documents, and you can understand why Gibbs wouldn’t want to work with her. But when Ari confesses to Kate’s death and other crimes in front of her and Gibbs, proving to Ziva that he’s too far gone, she does the right thing and shoots her half-brother before he can do the same to Gibbs. It’s probably the most difficult thing she ever has to do, and it haunts her for years after.
Honestly, you’d think that would be enough to prove her ability and allegiance, but no, not for the NCIS team. It isn’t until Ziva plays a key part in a case and saves another team member’s life that they finally accept her.