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