If $ a=e^{i\alpha}$ and $b=e^{i\beta} $ Show the following

Question

If $ a=e^{i\alpha}$ and $b=e^{i\beta} $, Show that $$ \frac{(a+b)(ab-1)}{(a-b)(ab+1)} = \frac{\sin(\alpha)+\sin(\beta)}{\sin(\alpha)-\sin(\beta)} $$

Answer 1

Hint

$$\dfrac{a^2b-b}{ab^2-a}=\cdots=\dfrac{ab\left(a-\dfrac1a\right)}{ab\left(b-\dfrac1b\right)}=\cdots=\dfrac{2i\sin\alpha}{?}$$

Now apply componendo dividendo