Question

Solve $\sin x+\sin y=\sin (x+y)$ and $|x|+|y|=1$.

Answer 1

$\begin{array}{c}Given\sin x+\sin y=\sin \left(x+y\right)\\ \Rightarrow 2\sin \frac{\left(x+y\right)}{2}\cos \frac{\left(x-y\right)}{2}\\ =2\sin \frac{\left(x+y\right)}{2}\cos \frac{\left(x+y\right)}{2}\\ \Rightarrow 2\sin \frac{\left(x+y\right)}{2}\left[\cos \left(\frac{x-y}{2}\right)-\cos \left(\frac{x+y}{2}\right)\right]=0\\ \Rightarrow 4\sin \left(\frac{x+y}{2}\right)\sin \frac{x}{2}\sin \frac{y}{2}=0\\ \sin \left(\frac{x+y}{2}\right)=0\Rightarrow x+y=0\\ \sin \frac{x}{2}=0\Rightarrow x=0\\ \sin \frac{y}{2}=0\Rightarrow y=0\end{array}$