Why didn't Annie kill Armin during the 57th Expedition Beyond the Walls? I watched the anime and couldn't find a explanation as to why Annie spared Armin's life?
Why would she?

Annie, at least in her mind, is not a villain. She may be an antagonist to the main characters in the Anime, but she does not think of herself as a villain. She does not think of herself as evil.

She would not benefit in the least from killing Armin. The only members of the expedition she killed were the ones that threatened her titan form. She trained alongside them, and even had formed bonds of friendship with some of them. Even the explorer corps members were at a discomfort dealing with her, after learning what she did.

Her main goal was to get Eren and deliver him to her masters/employers (it is still unclear who charged her with the kidnapping in the manga - as of chapter 66).

So, her reasons for not killing Armin may be:


Armin was harmless. His vertical maneuvering skills were poor, and even if he were to attack her titan form, she would easily squat him.
She does not think of herself as a monster, or even that she is evil, so she has zero motivation for random killing.
More on her self-image, she is clearly a human that can change into titan (in contrast to Ymir who was a titan for 60 years before gaining the power of shifting). She has a good life with the MP Brigade. She is not a psychopath.
She may have some feelings of friendship or camaraderie towards Armin. Remember Armin has a very likeable personality. He gets along well with everyone.

Regarding her personality, Shingeki no Kyojin Wiki says: 


  Yet, she somehow does seem to hold a strange fascination and even feelings of respect towards people that have a deep sense of duty and righteousness


And it seems to me that Armin falls under that category.
Maybe killing Armin at that time would take more time than just confirming he is not Eren, and she is in a hurry to complete her goal.
Armin was not alone. Killing him (remember that people actually LIKE Armin) could usher the other explorer corps around into a battle-frenzy to avenge his companion (maybe not likely, but who knows).


Reasons for killing Armin at that moment? Can't think of any.