$\dfrac{sin(a+b)}{sin(a-b)}\\=\dfrac{sin(a)cos(b)+cos(a)sin(b)}{sin(a)cos(b)-cos(a)sin(b)}\\=\dfrac{\dfrac{sin(a)cos(b)}{cos(a)cos(b)}+\dfrac{cos(a)sin(b)}{cos(a)cos(b)}}{\dfrac{sin(a)cos(b)}{cos(a)cos(b)}-\dfrac{cos(a)sin(b)}{cos(a)cos(b)}}\\=\dfrac{\dfrac{sin(a)}{cos(a)}+\dfrac{sin(b)}{cos(b)}}{\dfrac{sin(a)}{cos(a)}-\dfrac{sin(b)}{cos(b)}}\\=\dfrac{tan(a)+tan(b)}{tan(a)-tan(b)}$