\(\sin x = \frac{opp}{Hyp} = \frac{4}{5}\) 5\(^2\) = 4\(^2\) + adj\(^2\) adj\(^2\) = 25 - 16 = 9 adj = \(\sqrt{9}\) = 3 \(\tan x = \frac{4}{3}\) \(\cot x = \frac{1}{\frac{4}{3}} = \frac{3}{4}\) \(\cot^2 x = (\frac{3}{4})^2 = \frac{9}{16}\) \(\csc x = \frac{1}{\sin x}\) = \(\frac{1}{\frac{4}{5}} = \frac{5}{4}\) \(\csc^2 x = (\frac{5}{4})^2 = \frac{25}{16}\) \(\therefore \frac{1 + \cot^2 x}{\csc^2 x - 1} = \frac{1 + \frac{9}{16}}{\frac{25}{16} - 1}\) = \(\frac{25}{16} \div \frac{9}{16}\) = \(\frac{25}{9}\)