It is given that:-
[tex]\begin{gathered} \csc ^2\beta\times\cos (2\beta) \\ =\frac{1}{\sin^2\beta}\times(\cos ^2\beta-\sin ^2\beta) \\ =\frac{\cos^2\beta}{\sin^2\beta}-\frac{\sin ^2\beta}{\sin ^2\beta} \\ =\cot ^2\beta-1 \end{gathered}[/tex]So the correct answer is
[tex]\cot ^2\beta-1[/tex]