Is there an algorithm or preferred method to simplify these expressions?
$$\frac{\sin\alpha-\sin\beta}{\cos\alpha-\cos\beta}\quad\text{and}\quad \frac{\sin\alpha+\sin\beta}{\cos\alpha+\cos\beta}$$
I tried to manipulate them using double and sum but none of them seemed to help.
$$\frac{\sin\alpha-\sin\beta}{\cos\alpha-\cos\beta}=\frac{2\sin\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2}}{2\sin\frac{\beta-\alpha}{2}\sin\frac{\alpha+\beta}{2}}=-\cot\frac{\alpha+\beta}{2}.$$ The second is similar.