1. **Problem Statement:** Simplify the expression $\cot \alpha - \cot (\alpha + \beta)$. 2. **Recall the cotangent subtraction formula:** For any angles $x$ and $y$, $$\cot x - \cot y = \frac{\sin(y - x)}{\sin x \sin y}.$$ 3. **Apply the formula:** Let $x = \alpha$ and $y = \alpha + \beta$, then $$\cot \alpha - \cot (\alpha + \beta) = \frac{\sin((\alpha + \beta) - \alpha)}{\sin \alpha \sin (\alpha + \beta)} = \frac{\sin \beta}{\sin \alpha \sin (\alpha + \beta)}.$$ 4. **Interpretation:** The difference of cotangents simplifies to a ratio involving sine functions of the angles. This is useful in trigonometric simplifications and solving equations. **Final answer:** $$\cot \alpha - \cot (\alpha + \beta) = \frac{\sin \beta}{\sin \alpha \sin (\alpha + \beta)}.$$