 Solution

Apply the Itô formula separately to cosine and sine, using $[B{]}_t=t$:In vector standard form $d{Z}_t=\sigma (Z_t)d{B}_t+b(Z_t)dt$, with one-dimensional driving noise and two-dimensional state, the coefficient functions areThe original trigonometric process starts at $(1,0)$, but these coefficient functions define a stochastic differential equation on all of ${\mathbb{R}}^2$.