## Precalculus (6th Edition) Blitzer

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.