 Solution

The Itô formula applied to $F(t,x)=e^{t/2}\cos x$ gives exact cancellation of the drift:On every finite interval $[0,T]$, the Itô integral is square-integrable since $\mathbb{E}{\int }_0^T{e}^s{\sin }^2{B}_{s}ds\le e^T-1$. Therefore $e^{t/2}\cos B_t$ is a true martingale, and hence a local martingale.

For the other stochastic process, $d(B_t-t^2)=d{B}_t-2tdt$. If $B_t-t^2$ were a local martingale, its difference with $B$ would make the continuous finite-variation process $-t^2$ a local martingale, contradicting the fact that a continuous finite-variation local martingale is constant. Also $\mathbb{E}(B_t-t^2)=-t^2$ is not constant. Thus $B_t-t^2$ is neither a local martingale nor a martingale.