Answer
Please see below.
Work Step by Step
We verify as follows:
$$\csc \theta + \cot \theta =\frac{1}{\sin \theta }+ \frac{\cos \theta }{\sin \theta }=\frac{1+\cos \theta }{\sin \theta }=\frac{(1+\cos \theta )(1- \cos \theta )}{\sin \theta (1-\cos \theta ) }= \frac{1-\cos ^2 \theta }{\sin \theta (1-\cos \theta ) }=\frac{\sin ^2 \theta }{\sin \theta (1-\cos \theta ) }=\frac{\sin \theta }{1-\cos \theta }$$
