The number of pairs \((x, y)\) satisfying the equation \(\sin x + \sin y = \sin(x + y)\) and \(|x| + |y| = 1\) is

There are $6$ ordered pairs. The sine equation factors as $4\sin\frac{x}{2}\sin\frac{y}{2}\sin\frac{x+y}{2}=0$, and the bounds give $x=0$, $y=0$, or $x+y=0$. Each line meets $|x|+|y|=1$ at two distinct points, giving four axis points and two more when $x+y=0$. Counting just the axes gives $4$.