Answer
$\frac{\cos(\alpha-\beta)}{\cos\alpha\sin \beta}=\tan\alpha+\cot\beta$
Work Step by Step
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:
$=\frac{\cos\alpha\cos\beta}{\cos \alpha\sin\beta}+\frac{\sin\alpha\sin\beta}{\cos \alpha\sin\beta}$
$=\frac{\cos\beta}{\sin\beta}+\frac{\sin\alpha}{\cos \alpha}$
$=\cot\beta+\tan \alpha$
$=\tan\alpha+\cot\beta$
