Answer
Identity proved.
Work Step by Step
In order to prove the given identity, we simplify the left hand side $\text{LHS}$ as follows:
Multiply the numerator and denominator of the LHS by $\sin \ v$ to obtain:
$\text{LHS } =\dfrac{\sin v(\csc v -1) }{\sin v (\csc v +1)} \\= \dfrac{\csc v \sin v -\sin v }{\csc v \sin v +\sin v}$
Since, $\csc v \sin v =\dfrac{1}{\sin v } \cdot {\sin v}=1$
Therefore, $\dfrac{\csc v \sin v -\sin v }{\csc v \sin v +\sin v} = \dfrac{1-\sin
\ v}{ 1+\sin v } \\= \text{ RHS}$
