By the power and chain rules,
[tex]\dfrac{\mathrm dy}{\mathrm dx}=-(\cot x-\csc x)^{-2}\dfrac{\mathrm d(\cot x-\csc x)}{\mathrm dx}[/tex]
Then
[tex]\dfrac{\mathrm dy}{\mathrm dx}=-(\cot x-\csc x)^{-2}(-\csc^2x+\csc x\cot x)[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\csc^2x-\csc x\cot x}{(\cot x-\csc x)^2}[/tex]
Multiply through both the numerator and denominator by [tex]\sin^2x[/tex]:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{1-\cos x}{(\cos x-1)^2}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac{\cos x-1}{(\cos x-1)^2}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac1{\cos x-1}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac1{1-\cos x}[/tex]
