sin θ 1 - cot θ + cos θ 1 - tan θ =

$\frac{\sin \theta }{1-\cot \theta }+\frac{\cos {\displaystyle \theta }}{1-\tan \theta }=\frac{\sin {\displaystyle \theta }}{1-{\displaystyle \frac{\cos \theta }{\sin \theta }}}+\frac{\cos {\displaystyle \theta }}{1-{\displaystyle \frac{\sin \theta }{\cos \theta }}}$

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

$=\frac{{\sin }^2\theta -{\cos }^2\theta }{\sin \theta -\cos \theta }=\frac{(\sin \theta +\cos \theta )(\sin \theta -\cos \theta )}{(\sin \theta -\cos \theta )}$

$=\sin \theta +\cos \theta$.