sin(x+y)/(sin(x)+sin(y)=(sin(x)-sin(y))/sin(x-y)

\frac{\sin (x+y)}{\sin x+ \sin y} = \frac{\sin x - sin y}{\sin (x-y)} :

\sin (x+y) \sin (x-y) = (\sin x + \sin y) (\sin x - \sin y)

\sin (x+y) \sin (x-y) =
(\sin x\cos y + \sin y\cos x)(\sin x\cos y - \sin y \cos x)=
\sin^2 x\cos^2y - \sin^2y\cos^2x =
\sin^2x(1-\sin^2y)-\sin^2y(1-\sin^2 x)=
\sin^2x - \sin^2x\sin^2y - \sin^2y + \sin^2x\sin^y=
\sin^2x-\sin^2y

(\sin x + \sin y) (\sin x - \sin y)=\sin^2x-\sin^2y

Thanks this was a big help !