Thread: Cos of Brownian motion

If you take the cosine of brownian motion, can you make the assumption it's 1?
As brownian motion has a mean of zero, over time taking the cos at each time step it'll average out to one?
sorry if silly question, but i want to make the assumption in a problem that (1- cos x).dt (x being brownian) this term for all intents and purposes is zero.

ah i didn't do ito properly in the previous step. ignore this question.