Let's simplify the left side and see if it matches the right side:

Given: $\frac{\cos(\alpha + \beta)}{\cos \alpha \sin \beta} = \cot \beta - \tan \alpha$

Step 1: Expand $\cos(\alpha + \beta)$

Using the angle addition formula for cosine: $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ Thus, the left side becomes: $\frac{\cos \alpha \cos \beta - \sin \alpha \sin \beta}{\cos \alpha \sin \beta}$

Step 2: Separate the Fraction

Now, split the fraction into two parts: $= \frac{\cos \alpha \cos \beta}{\cos \alpha \sin \beta} - \frac{\sin \alpha \sin \beta}{\cos \alpha \sin \beta}$

Step 3: Simplify Each Term

Simplify each part of the fraction separately:

1. $\frac{\cos \alpha \cos \beta}{\cos \alpha \sin \beta} = \frac{\cos \beta}{\sin \beta} = \cot \beta$
2. $\frac{\sin \alpha \sin \beta}{\cos \alpha \sin \beta} = \frac{\sin \alpha}{\cos \alpha} = \tan \alpha$

Thus, the left side simplifies to: $\cot \beta - \tan \alpha$

Conclusion

Since the left side simplifies to the right side, the given identity is true: $\frac{\cos(\alpha + \beta)}{\cos \alpha \sin \beta} = \cot \beta - \tan \alpha$

Would you like further details on any part of this derivation?

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Here are 5 related questions to explore further:

1. How would this identity change if we used $\sin(\alpha + \beta)$ instead of $\cos(\alpha + \beta)$?
2. What are other identities involving $\tan$ and $\cot$ that might be useful in trigonometric proofs?
3. Can we derive similar identities for $\sin(\alpha + \beta)$ or $\tan(\alpha + \beta)$?
4. How would we approach proving identities that involve $\sec$ and $\csc$?
5. Are there applications of this identity in solving real-world problems or geometry?

Tip: Remember, breaking down complex identities by separating fractions often makes simplification much easier!