Step-by-step Solution:

Since \[ \csc^2 \theta - \cot^2 \theta = 1 \Rightarrow \csc \theta - \cot \theta = \frac{1}{\csc \theta + \cot \theta}, \] Hence, \[ \csc \theta - \cot \theta = \frac{1}{2}. \] As we are given that \[ \csc \theta - \cot \theta = 2, \] Adding the two equations, we have: \[ 2 \csc \theta = 2 + \frac{1}{2} \Rightarrow \csc \theta = \frac{5}{4}. \]