Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2008 / A / 3 / b / Solution

For $f(z)=\sin z$, $f^{\prime }(z)=\cos z$ and $f^{\prime \prime }(z)=-\sin z$. Itô's formula givesSince $B_0=0$, this is exactlyThe integrand retains $\cos B_t$; replacing it by a fixed sign of $\sqrt{1-X_t^2}$ would lose information about the phase of $B_t$.