= Solution
Apply the two-variable <Itô formula> to $e^w\cos b$. The independent <Brownian motions> satisfy $[B,W]=0$, while the two diagonal second derivatives sum to zero. Consequently
$$
dY_t=e^{W_t}\cos B_t\,dW_t-e^{W_t}\sin B_t\,dB_t,\qquad [Y]_t=\int_0^te^{2W_s}\,ds.
$$
Thus $Y$ is a <continuous local martingale>, but \b[it is not Brownian motion]. Brownian motion from any fixed starting point has quadratic variation $t$. If the displayed bracket were identically $t$, differentiating its continuous density would force $e^{2W_t}=1$ for every $t$, hence $W_t=0$ identically, an event of probability zero. This is the <cosine-exponential Brownian local martingale>. Its initial value $Y_0=1$ is retained in the next part.
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