AMC 12 2018a Problem 9

Which of the following describes the largest subset of values of $y$ within the closed interval $[0,\pi]$ for which $$\sin(x+y)\leq \sin(x)+\sin(y)$$for every $x$ between $0$ and $\pi$, inclusive?

$\textbf{(A) } y=0 \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \textbf{(E) } 0\leq y\leq \pi$