Product of orthogonal martingales

Question

Let $(\Omega,\mathcal F,\mathcal F_t,P)$ be a probability space and $W(t)$ be a Brwonian Motion defined on it. Let $M(t)$ be a bounded martingale orthogonal to $W(t)$, i.e., there exists a constant $C>0$ such that $\sup_{t,\omega}|M(t)|\leq C$ and $M(t)W(t)$ is a martingale. For $t\geq s$, is the following ture: $$E[(M(t)-M(s))cos(W(t)-W(s))|\mathcal F_s]=0?$$ Thanks!