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