Answer
Refer to the proof below.
Work Step by Step
In order to prove the given identity, we simplify the left hand side $\text{LHS}$.
We know that $\sin \theta =\dfrac{1}{\csc \theta}$
$\text{LHS } =\dfrac{1+\dfrac{1}{\csc \theta }}{1-\dfrac{1}{\csc \theta} } \\=\dfrac{\dfrac{\csc \theta +1}{\csc \theta}}{\dfrac{\csc \theta -1}{\csc \theta}} \\= \dfrac{\csc \theta +1}{\csc \theta -1}\\= \\= \text{ RHS}$
Thus the left-hand side equals the right-hand side and we have proven the identity.
