Solution: \begin{aligned} \frac{1+\sin \theta-\cos \theta}{1+\sin \theta+\cos \theta} & \ = \frac{1-\cos \theta+\sin \theta}{1+\cos \theta+\sin \theta} \ & = \frac{2 \sin ^{2} \frac{\theta}{2}+2 \sin \frac{\theta}{2} \times \cos \frac{\theta}{2}}{2 \cos ^{2} \frac{\theta}{2}+2 \sin \frac{\theta}{2} \times \cos \frac{\theta}{2}} \ & =\frac{2 \sin \frac{\theta}{2}\left(\sin \frac{\theta}{2}+\cos \frac{\theta}{2}\right)}{2 \cos \frac{\theta}{2}\left(\sin \frac{\theta}{2}+\cos \frac{\theta}{2}\right)} \ & = \tan \left(\frac{\theta}{2}\right) \end{aligned}