The given mathematical expression is:

$\frac{\tan \alpha - \cot \beta}{\cot \alpha - \tan \beta} = - \tan \alpha \cot \beta$

This equation seems to deal with trigonometric functions, particularly involving the tangent ($\tan$) and cotangent ($\cot$) of two angles, $\alpha$ and $\beta$. The left-hand side (LHS) is a fraction where the numerator is $\tan \alpha - \cot \beta$ and the denominator is $\cot \alpha - \tan \beta$. The right-hand side (RHS) is $-\tan \alpha \cot \beta$.

Steps to Simplify or Prove:

1. You can attempt to express all the terms in terms of sine and cosine functions to simplify or prove the identity. Recall the identities: $\tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta}$

   This may help to manipulate both sides of the equation.

2. Another approach could involve multiplying both sides of the equation by $\sin \alpha \sin \beta$ or $\cos \alpha \cos \beta$ to simplify the terms further.

Would you like me to proceed with a detailed simplification or proof of this trigonometric identity?

Here are five related questions for deeper understanding:

1. How do the tangent and cotangent functions relate to sine and cosine?
2. What are some standard methods for simplifying trigonometric identities?
3. How can multiplying both sides of a trigonometric equation help in proving identities?
4. What are some common tricks for handling complex trigonometric expressions?
5. Why are tangent and cotangent often paired in trigonometric identities?

Tip: When simplifying trigonometric identities, converting all terms to sine and cosine often reveals patterns that simplify complex expressions.