Let $$\begin{cases} dX_t = \mathrm{sin}(X_t+Y_t) dW_t \\ dY_t = \mathrm{cos}(X_t+Y_t) dV_t \\ X_0=Y_0=0 \end{cases}$$
Where $(W,V)$ is a two-dimensional Brownian motion and $(X,Y)$ be a strong solution to the set of equations above. Find $a,b \in \mathbb{R}$, so that $aX_t+bY_t$ is a Brownian motion as well.
I tried to prove it as shown in this question: Stochastic integral of $\mathrm{sin}(X_t+Y_t)\mathrm{cos}(X_t+Y_t)$ which required to compute the stochastic integral of cosine and sine but from the comments it follows that there is another way of proving it, which I would be grateful for showing.