LHS
$\large\frac{1+\sin \theta-\cos \theta}{1+\sin \theta+\cos \theta}$
$\large\frac{1-\cos \theta+\sin \theta}{1+\cos \theta+\sin \theta}=\frac{2\sin ^2\Large\frac{\theta}{2}+2\sin \Large\frac{\theta}{2}.\cos\large\frac{\theta}{2}}{2\cos ^2\Large\frac{\theta}{2}+2\sin \Large\frac{\theta}{2}.\cos\large\frac{\theta}{2}}$
$\Rightarrow \large\frac{2\sin \large\frac{\theta}{2}(\sin \Large\frac{\theta}{2}+\cos\Large\frac{\theta}{2})}{2\sin \large\frac{\theta}{2}(\sin \Large\frac{\theta}{2}+\cos\Large\frac{\theta}{2})}$
$\Rightarrow \tan\large\frac{\theta}{2}$
Hence proved