Answer
$$\sin \theta + \cos \theta = \frac{{\sin \theta }}{{1 - \cot \theta }} + \frac{{\cos \theta }}{{1 - \tan \theta }}$$
Work Step by Step
$$\eqalign{
& \sin \theta + \cos \theta = \frac{{\sin \theta }}{{1 - \cot \theta }} + \frac{{\cos \theta }}{{1 - \tan \theta }} \cr
& {\text{We transform the more complicated right side to match the left side}}. \cr
& \frac{{\sin \theta }}{{1 - \cot \theta }} + \frac{{\cos \theta }}{{1 - \tan \theta }} = \frac{{\sin \theta }}{{1 - \cot \theta }}\left( {\frac{{\sin \theta }}{{\sin \theta }}} \right) + \frac{{\cos \theta }}{{1 - \tan \theta }}\left( {\frac{{\cos \theta }}{{\cos \theta }}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\sin }^2}\theta }}{{\sin \theta - \cot \theta \sin \theta }} + \frac{{{{\cos }^2}\theta }}{{1 - \tan \theta \cos \theta }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\sin }^2}\theta }}{{\sin \theta - \cot \theta \sin \theta }} + \frac{{{{\cos }^2}\theta }}{{\cos \theta - \tan \theta \cos \theta }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\sin }^2}\theta }}{{\sin \theta - \cos \theta }} + \frac{{{{\cos }^2}\theta }}{{\cos \theta - \sin \theta }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\sin }^2}\theta }}{{\sin \theta - \cos \theta }} - \frac{{{{\cos }^2}\theta }}{{\sin \theta - \cos \theta }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{\sin }^2}\theta - {{\cos }^2}\theta }}{{\sin \theta - \cos \theta }} \cr
& {\text{ Factoring the numerator}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{\left( {\sin \theta - \cos \theta } \right)\left( {\sin \theta + \cos \theta } \right)}}{{\sin \theta - \cos \theta }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \sin \theta + \cos \theta \, \cr
& {\text{Thus have verified that the given equation is an identity}}\,\, \cr} $$
