a)
[tex]\sin x+ \sin y - \sin(x+y) = 2\sin \dfrac{x+y}{2}\cos \dfrac{x-y}{2}-\sin\Big(2\cdot \dfrac{x+y}{2}\Big) = \\ \\ = 2\sin \dfrac{x+y}{2}\cos \dfrac{x-y}{2}- 2\sin \dfrac{x+y}{2}\cos \dfrac{x+y}{2} = \\ \\ = 2\sin \dfrac{x+y}{2}\Big(\cos \dfrac{x-y}{2} - \cos \dfrac{x+y}{2} \Big) = \\ \\ = 2\sin\dfrac{x+y}{2} \Big(2\sin\dfrac{(x+y)-(x-y)}{2}\sin \dfrac{(x+y)+(x-y)}{2}\Big) = \\ \\ = 2\sin\dfrac{x+y}{2} \cdot 2\sin \dfrac{y}{2}\sin \dfrac{x}{2} = \\ \\ = 4\sin\dfrac{x}{2}\sin \dfrac{y}{2}\sin \dfrac{x+y}{2}[/tex]
