Answer
See proof below.
Work Step by Step
In order to prove the given identity, we simplify the left hand side $\text{LHS}$.
Multiply by the numerator and denominator of the LHS by $\csc \theta +1$ to obtain:
$\text{LHS } =\dfrac{\csc \theta -1}{\cot \theta } \cdot \dfrac{\csc \theta +1}{\csc \theta +1} \\=\dfrac{\csc^2 \theta -1}{\cot \theta (\csc \theta +1)} \\= \dfrac{\cot^2 \theta}{\cot \theta (\csc \theta +1)}\\= \dfrac{\cot \theta}{(\csc \theta +1)} \\= \text{ RHS}$
