Official solution
Given expression can be written as
\[{{\sin A} \over {\cos A}} \times {{sin\,A} \over {\sin A - \cos A}} + {{\cos A} \over {\sin A}} \times {{\cos A} \over {\cos A - sin\,A}}\]
(As \[\tan A = {{\sin A} \over {\cos A}}\] and \[\cot A = {{\cos A} \over {\sin A}}\] )
\[ = {1 \over {\sin A - \cos A}}\left\{ {{{{{\sin }^3}A - {{\cos }^3}A} \over {\cos A\sin A}}} \right\}\]
\[ = {{{{\sin }^2}A + \sin A\cos A + {{\cos }^2}\,A} \over {\sin A\cos A}}\]
\[ = 1 + \sec\, A{\mathop{\rm cosec}\nolimits} \,A\]