The number of pairs $(x, y)$ satisfying the equations $\sin x + \sin y = \sin(x + y)$ and $|x| + |y| = 1$ is:
Step-by-Step Solution
Key Concept: Product-to-sum identities transform the given equation into a form where each factor can independently equal zero.
From $\sin x + \sin y = \sin(x+y)$, we use sum-to-product formulas to obtain $4\sin\frac{x+y}{2}\sin\frac{x}{2}\sin\frac{y}{2} = 0$, giving either $\sin\frac{x+y}{2} = 0$, $\sin\frac{x}{2} = 0$, or $\sin\frac{y}{2} = 0$. Combined with constraint $|x| + |y| = 1$, we find solutions: $(0, \pm 1)$, $(\pm 1, 0)$, $(\frac{1}{2}, -\frac{1}{2})$, and $(-\frac{1}{2}, \frac{1}{2})$.
Correct Answer: 4