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 $\cot A = {{\cos A} \over {\sin A}}$ ) $= {1 \over {\sin A - \cos A}}\left\{ {{{{{\sin }^3}A - {{\cos }^3}A} \over {\cos A\sin A}}} \right\}$ $= {{{{\sin }^2}A + \sin A\cos A + {{\cos }^2}\,A} \over {\sin A\cos A}}$ $= 1 + \sec\, A{\mathop{\rm cosec}\nolimits} \,A$