For any real numbers x and y. Sinx = siny implies $\displaystyle n\pi + (-1)^ny, where n \in Z$

If sinx = siny, then sinx -siny = 0

or$\displaystyle 2cos\frac{x+y}{2}sin\frac{x-y}{2} = 0$

$\displaystyle \Rightarrow cos\frac{x+y}{2} = 0 ; sin\frac{x-y}{2}=0$

Therefore $\displaystyle \frac{x+y}{2} = (2n+1)\frac{\pi}{2} or \frac{x-y}{2} = n\pi,$ where $\displaystyle n \in Z$

i.e. X = $\displaystyle (2n+1)\pi -y or x = 2n\pi +y$, where $\displaystyle n \in Z$

Please guide further.....