= Solution
Apply the <Itô formula> separately to cosine and sine, using $[B]_t=t$:
$$
\boxed{dX_t=-Y_t\,dB_t-\tfrac12X_t\,dt,\qquad dY_t=X_t\,dB_t-\tfrac12Y_t\,dt.}
$$
In vector standard form $dZ_t=\sigma(Z_t)\,dB_t+b(Z_t)\,dt$, with one-dimensional driving noise and two-dimensional state, the coefficient functions are
$$
\boxed{\sigma(x,y)=\begin{pmatrix}-y\\x\end{pmatrix},\qquad b(x,y)=-\tfrac12\begin{pmatrix}x\\y\end{pmatrix}.}
$$
The original trigonometric process starts at $(1,0)$, but these coefficient functions define a <stochastic differential equation> on all of $\mathbb R^2$.
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