$$\frac{1+\cos\theta+\sin\theta}{1+\sin\theta – \cos\theta} \;=\; \frac{1 + \cos\theta}{\sin\theta}$$
Note: The smaller blue hypotenuse bisects a red angle of measure \(\theta\).
(Why?) From this, we may also deduce:
$$\frac{\sin\theta}{1+\cos\theta} \;=\; \tan\frac{\theta}{2}$$