algebra precalculus - Continuous function and function relationship

I have a function relationship and I know that this function is continuous in a specific $x_0$. I know how to prove that this function is continuous at $D_f$.

I have the function relationship:
$$f(x+y)=f(x)\cdot f(y)-\sin(x)\cdot \sin(y), \forall x,y\in\mathbb{R}$$ This function is continuous in $x_0=0$ and I want to prove that $f$ is continuous at $\mathbb{R}$. But for $x=y=0$ I get two different values for $f(0)$. So how to prove what I want?