Answer
$$\frac{{1 + \cos x}}{{1 - \cos x}} - \frac{{1 - \cos x}}{{1 + \cos x}} = 4\cot x\csc x$$
Work Step by Step
$$\eqalign{
& \frac{{1 + \cos x}}{{1 - \cos x}} - \frac{{1 - \cos x}}{{1 + \cos x}} = 4\cot x\csc x \cr
& {\text{We transform the more complicated left side to match the right side}}. \cr
& \frac{{1 + \cos x}}{{1 - \cos x}} - \frac{{1 - \cos x}}{{1 + \cos x}} = \frac{{{{\left( {1 + \cos x} \right)}^2} - {{\left( {1 - \cos x} \right)}^2}}}{{\left( {1 - \cos x} \right)\left( {1 + \cos x} \right)}} \cr
& \frac{{1 + \cos x}}{{1 - \cos x}} - \frac{{1 - \cos x}}{{1 + \cos x}} = \frac{{1 + 2\cos x + {{\cos }^2}x - 1 + 2\cos x - {{\cos }^2}x}}{{1 - {{\cos }^2}x}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{4\cos x}}{{{{\sin }^2}x}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4\left( {\frac{{\cos x}}{{\sin x}}} \right)\left( {\frac{1}{{\sin x}}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4\cot x\csc x \cr
& {\text{Thus have verified that the given equation is an identity}} \cr
& 76************************************************* \cr
& \frac{{1 + \sin \theta }}{{1 - \sin \theta }} - \frac{{1 - \sin \theta }}{{1 + \sin \theta }} = 4\tan \theta \sec \theta \cr
& {\text{We transform the more complicated left side to match the right side}}. \cr
& \frac{{1 + \sin \theta }}{{1 - \sin \theta }} - \frac{{1 - \sin \theta }}{{1 + \sin \theta }} = \frac{{{{\left( {1 + \sin \theta } \right)}^2} - {{\left( {1 - \sin \theta } \right)}^2}}}{{1 - {{\sin }^2}\theta }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{1 + 2\sin \theta + {{\sin }^2}\theta - 1 + 2\sin \theta - {{\sin }^2}\theta }}{{{{\cos }^2}\theta }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{4\sin \theta }}{{{{\cos }^2}\theta }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4\left( {\frac{{\sin \theta }}{{\cos \theta }}} \right)\left( {\frac{1}{{\cos \theta }}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4\tan \theta \sec \theta \cr
& {\text{Thus have verified that the given equation is an identity}} \cr} $$
