Let $(\Omega, \mathcal F, P)$ be a probability space and $\{B_t\}_{t\ge0}$ a Brownian motion. Furthermore let $\{F_t\}_{t\ge0}$ be the natural filtration of $B$. Let
$$Y(t)=\sin(t+B_t), \ \ \ t\ge0$$
I want to determine $dY(t)$ using Ito's lemma.
As we didn't use this lemma yet I don't really know how to solve this problem. I managed to determine $dY(t)$ for $Y_t=\sin(B_t)$ and for $Y_t=\cos(B_t)$.So I thought about using $\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$ and applying these results somehow but I don't think that's the right way to solve this.
$dY(t)=\frac{\partial Y}{\partial t}dt+\frac{\partial Y}{\partial B_t}dB_t+\frac{1}{2}\frac{\partial^2Y}{\partial B_t^2}(dB_t)^2=\cos(t+B_t)dt+\cos(t+B_t)dB_t-\frac{1}{2}\sin(t+B_t)dt$