You can use the sum of angles identities, then rearrange to put the result in the form of tangents.
[tex]\displaystyle\frac{\sin{(x+y)}}{\sin{(x-y)}}=\frac{\sin{(x)}\cos{(y)}+\cos{(x)}\sin{(y)}}{\sin{(x)}\cos{(y)}-\cos{(x)}\sin{(y)}}\\\\=\frac{\left(\frac{\sin{(x)}\cos{(y)}+\cos{(x)}\sin{(y)}}{\cos{(x)}\cos{(y)}}\right)}{\left(\frac{\sin{(x)}\cos{(y)}-\cos{(x)}\sin{(y)}}{\cos{(x)}\cos{(y)}}\right)}\\\\=\frac{\tan{(x)}+\tan{(y)}}{\tan{(x)}-\tan{(y)}}[/tex]
