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,
$\begin{align}
& \frac{\cos \left( \alpha -\beta \right)}{\sin \alpha \sin \beta }=\frac{\cos \alpha \cos \beta +\sin \alpha \sin \beta }{\sin \alpha \sin \beta } \\
& =\frac{\cos \alpha \cos \beta }{\sin \alpha \sin \beta }+\frac{\sin \alpha \sin \beta }{\sin \alpha \sin \beta } \\
& =\frac{\cos \alpha }{\sin \alpha }\frac{\cos \beta }{\sin \beta }+1
\end{align}$
Substitute $\frac{\cos \alpha }{\sin \alpha }=\cot \alpha \text{ and }\frac{\cos \beta }{\sin \beta }=\cot \beta $.
$\frac{\cos \left( \alpha -\beta \right)}{\sin \alpha \sin \beta }=\cot \alpha \cot \beta +1$
Since the left-hand side part of the identity is equivalent to the expression on the right-hand side, therefore, the identity is verified.
