\(\csc x = \frac{1}{\sin x}\)

Using the quotient rule,

\(\frac{\mathrm d y}{\mathrm d x} = \frac{vdu - udv}{v^{2}}\)

= \(\frac{\sin x (0) - 1 (\cos x)}{(\sin x)^{2}}\)

= \(\frac{- \cos x}{\sin^{2} x}\)

= \((\frac{- \cos x}{\sin x}) (\frac{1}{\sin x})\)

= \(- \cot x \csc x\)