Answer
See the full explanation below.
Work Step by Step
Evaluate the term $\cos \left( \alpha -\beta \right)$ using the cosines difference formula and solve the expression on the left-hand side of the identity as,
$\begin{align}
& \frac{\cos \left( \alpha -\beta \right)}{\cos \alpha \sin \beta }=\frac{\cos \alpha \cos \beta +\sin \alpha \sin \beta }{\cos \alpha \sin \beta } \\
& =\frac{\cos \alpha \cos \beta }{\cos \alpha \sin \beta }+\frac{\sin \alpha \sin \beta }{\cos \alpha \sin \beta } \\
& =\frac{\cos \beta }{\sin \beta }+\frac{\sin \alpha }{\cos \alpha }
\end{align}$
Substitute $\frac{\sin \alpha }{\cos \alpha }=\tan \alpha \text{ and }\frac{\cos \beta }{\sin \beta }=\cot \beta $.
$\frac{\cos \left( \alpha -\beta \right)}{\cos \alpha \sin \beta }=\tan \beta +\tan \alpha $
Since the left-hand side part of the identity is equivalent to the expression on the right-hand side, therefore, the identity is verified.
