We are tasked with proving the trigonometric identity:

$\sec x - \cos x + \csc x - \sin x - \sin x \tan x = \cos x \cot x$

Step-by-step Proof:

1. Rewrite in terms of basic trigonometric functions:

   • $\sec x = \frac{1}{\cos x}$,
   • $\csc x = \frac{1}{\sin x}$,
   • $\tan x = \frac{\sin x}{\cos x}$,
   • $\cot x = \frac{\cos x}{\sin x}$.

   Substituting these identities into the left-hand side (LHS) expression:

   $\frac{1}{\cos x} - \cos x + \frac{1}{\sin x} - \sin x - \sin x \cdot \frac{\sin x}{\cos x}$

2. Simplify the expression: Simplifying each term:

   • The first term remains $\frac{1}{\cos x}$,
   • The second term remains $- \cos x$,
   • The third term remains $\frac{1}{\sin x}$,
   • The fourth term remains $- \sin x$,
   • The fifth term becomes $- \frac{\sin^2 x}{\cos x}$.

   Now the LHS becomes:

   $\frac{1}{\cos x} - \cos x + \frac{1}{\sin x} - \sin x - \frac{\sin^2 x}{\cos x}$

3. Combine like terms: Group terms with $\frac{1}{\cos x}$ and $\frac{\sin^2 x}{\cos x}$:

   $\left( \frac{1}{\cos x} - \frac{\sin^2 x}{\cos x} \right) - \cos x + \frac{1}{\sin x} - \sin x$

   Simplify the expression inside the parentheses:

   $\frac{1 - \sin^2 x}{\cos x}$

   Using the Pythagorean identity $1 - \sin^2 x = \cos^2 x$, we simplify it further:

   $\frac{\cos^2 x}{\cos x} = \cos x$

   Thus, the LHS now simplifies to:

   $\cos x - \cos x + \frac{1}{\sin x} - \sin x$

4. Further simplification: The terms $\cos x - \cos x$ cancel out, leaving us with:

   $\frac{1}{\sin x} - \sin x$

5. Rewrite the right-hand side (RHS): The RHS is:

   $\cos x \cot x = \cos x \cdot \frac{\cos x}{\sin x} = \frac{\cos^2 x}{\sin x}$

6. Equate LHS and RHS: From step 4, the simplified LHS is $\frac{1}{\sin x} - \sin x$. However, it is clear that this form does not match the current simplified RHS $\frac{\cos^2 x}{\sin x}$.

   After verifying each term in detail, we find that either a correction is needed in the given equation (as some cancellation or special identity is possibly missing). Would you like further analysis on this discrepancy, or we can revisit the equation for possible re-arrangement?

Let me know if you need further exploration or explanations for specific parts of the equation.