When Daisy tells Nick, "Sophisticated--- God, i'm sophisticated" how is she hoping to be viewed? Why does Nick recognize a "basic insincerity?"
I think that when Daisy says this to Nick (in Chapter 1), she is trying to convince Nick that she is happy and in a good place in her life. She wants him to think that her life has turned out well.
Nick does not tell us what it is that makes him think that she is not telling the truth. He says he just can tell from the point that "her voice broke off" that she is not being sincere.
To me, it would be pretty weird to say to someone that you are really sophisticated. It seems like one of those things that you let people notice for themselves. So I think maybe he realizes she's insincere because she feels the need to say that. It's like someone telling you how smart they are or how good at sports -- it seems like a cover up for their real feelings.