1. **State the problem:** Prove or simplify the expression $$\frac{\cos(u - v)}{\cos u \sin v} = \tan u + \cot v$$. 2. **Recall the cosine difference formula:** $$\cos(u - v) = \cos u \cos v + \sin u \sin v$$ 3. **Substitute the formula into the left side:** $$\frac{\cos u \cos v + \sin u \sin v}{\cos u \sin v}$$ 4. **Split the fraction into two parts:** $$\frac{\cos u \cos v}{\cos u \sin v} + \frac{\sin u \sin v}{\cos u \sin v}$$ 5. **Simplify each term by canceling common factors:** $$\frac{\cancel{\cos u} \cos v}{\cancel{\cos u} \sin v} + \frac{\sin u \cancel{\sin v}}{\cos u \cancel{\sin v}} = \frac{\cos v}{\sin v} + \frac{\sin u}{\cos u}$$ 6. **Recognize the trigonometric functions:** $$\frac{\cos v}{\sin v} = \cot v \quad \text{and} \quad \frac{\sin u}{\cos u} = \tan u$$ 7. **Rewrite the expression:** $$\tan u + \cot v$$ **Final answer:** $$\frac{\cos(u - v)}{\cos u \sin v} = \tan u + \cot v$$ This confirms the given identity.