## Precalculus (6th Edition)

$\frac{\cos(\alpha-\beta)}{\cos\alpha\sin \beta}=\tan\alpha+\cot\beta$
Start with the left side: $\frac{\cos(\alpha-\beta)}{\cos\alpha\sin \beta}$ Use the subtraction formula for cosine: $=\frac{\cos\alpha\cos\beta+\sin\alpha\sin\beta}{\cos \alpha\sin\beta}$ Break it into two fractions and simplify: $=\frac{\cos\alpha\cos\beta}{\cos \alpha\sin\beta}+\frac{\sin\alpha\sin\beta}{\cos \alpha\sin\beta}$ $=\frac{\cos\beta}{\sin\beta}+\frac{\sin\alpha}{\cos \alpha}$ $=\cot\beta+\tan \alpha$ $=\tan\alpha+\cot\beta$