Given the expression;
[tex]\csc \theta-\csc \theta\cdot\cos ^2\theta[/tex]This can be simplified as;
[tex]\csc \theta-\csc \theta\cdot\cos ^2\theta=\csc \theta(1-\cos ^2\theta)[/tex]Recall the identity that;
[tex]\begin{gathered} \sin ^2\theta+\cos ^2\theta=1 \\ \sin ^2\theta=1-\cos ^2\theta \end{gathered}[/tex]Then, we have;
[tex]\csc \theta-\csc \theta\cdot\cos ^2\theta=\csc \theta(\sin ^2\theta)[/tex]Also, recall that;
[tex]\csc \theta=\frac{1}{\sin \theta}[/tex]So, we have;
[tex]\begin{gathered} \csc \theta-\csc \theta\cdot\cos ^2\theta=\csc \theta(\sin ^2\theta) \\ \csc \theta-\csc \theta\cdot\cos ^2\theta=\frac{1}{\sin \theta}(\sin ^2\theta) \\ \csc \theta-\csc \theta\cdot\cos ^2\theta=\sin \theta \end{gathered}[/tex]CORRECT OPTION: D
