Proving $\frac{\cos a - \sin a + 1}{\cos a + \sin a - 1 } = \frac{\sin a}{1-\cos a}$

Question

Can somebody help to prove that:

$$\frac{\cos a - \sin a + 1}{\cos a + \sin a - 1 } = \frac{\sin a}{1-\cos a}$$

Answer 1

\begin{eqnarray*} \frac{ \cos a -\sin a +1}{\cos a +\sin a -1} \times \frac{1-\cos a}{1-\cos a} =\frac{\color{red}{\cos a}-\sin a+\overbrace{1-\cos^2 a}^{\sin^2 a}+\sin a \cos a\color{red}{-\cos a}}{(\cos a +\sin a -1)((1-\cos a)} =\frac{\sin a (\cos a +\sin a -1)}{(1-\cos a)(\cos a +\sin a -1)} \end{eqnarray*}