$\frac{1-\cos 2\theta +\sin 2\theta }{1+\cos 2\theta +\sin 2\theta }=\frac{1-\left(1-2{\sin }^2\theta \right)+2\sin \theta .\cos \theta }{1+\left(2{\cos }^2\theta -1\right)+2\sin \theta .\cos \theta }$

(Using formula $\cos 2\theta =1-2{\sin }^2\theta =2{\cos }^2\theta -1 \& \sin 2\theta =2\sin \theta .\cos \theta$)

$=\frac{\cancel{1}-\cancel{1}+2{\sin }^2\theta +2\sin \theta .\cos \theta }{\cancel{1}+2{\cos }^2\theta -\cancel{1}+2\sin \theta .\cos \theta }$

$=\frac{2{\sin }^2\theta +2\sin \theta .\cos \theta }{2{\cos }^2\theta +2\sin \theta .\cos \theta }$

$=\frac{2\sin \theta \cancel{\left(\sin \theta +\cos \theta \right)}}{2\cos \theta \cancel{\left(\sin \theta +\cos \theta \right)}}=\tan \theta$