Brownian sine terminal martingale ($\sin B_T={\int }_0^T{e}^{-(T-s)/2}\cos B_{s}d{B}_s$)

The conditional expectation of $\sin B_T$ at time $t$ is $e^{-(T-t)/2}\sin B_t$, by independent Brownian increment. Itô's formula cancels its time derivative against its second spatial derivative and leaves the displayed stochastic integral. The integrand is bounded, so Itô isometry verifies its square integrability.