Question

Find the derivative of the function y = (cot x- csc x)^-1
dy/dx = ?

Answer 1

By the power and chain rules,

`(\mathrm dy)/(\mathrm dx)=-(\cot x-\csc x)^(-2)(\mathrm d(\cot x-\csc x))/(\mathrm dx)`

Then

`(\mathrm dy)/(\mathrm dx)=-(\cot x-\csc x)^(-2)(-\csc^2x+\csc x\cot x)`

`(\mathrm dy)/(\mathrm dx)=(\csc^2x-\csc x\cot x)/((\cot x-\csc x)^2)`

Multiply through both the numerator and denominator by
`\sin^2x`:

`(\mathrm dy)/(\mathrm dx)=(1-\cos x)/((\cos x-1)^2)`

`(\mathrm dy)/(\mathrm dx)=-(\cos x-1)/((\cos x-1)^2)`

`(\mathrm dy)/(\mathrm dx)=-\frac1{\cos x-1}`

`(\mathrm dy)/(\mathrm dx)=\frac1{1-\cos x}`