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