Question

Verify the identity.

cotangent of x divided by quantity one plus cosecant of x equals quantity cosecant of x minus one divided by cotangent of x

Answer 1

Identity to verify:

`(\cot x)/(1+\csc x)=(\csc x-1)/(\cot x)`

Recall that

`\cos^2x+\sin^2x=1`

Divide both sides by
`\sin^2x` and we get

`\cot^2x+1=\csc^2x`

or

`\cot^2x=\csc^2x-1=(\csc x-1)(\csc x+1)`

So if we multiply the numerator and denominator of

`(\cot x)/(1+\csc x)`

by
`\csc x-1`, we get

`(\cot x(\csc x-1))/((1+\csc x)(\csc x-1))=(\cot x(\csc x-1))/(\csc^2x-1)=(\cot x(\csc x-1))/(\cot^2x)`

Then as long as
`\cot x\\eq0`, we can cancel terms to end up with

`(\csc x-1)/(\cot x)`

and establish the identity.