Question

How do you verify the identity `(csc x - cot x)^2 = (1 - cos x)/(1+cosx)`?

Answer 1

See below

Explanation:

`(csc-cotx)^2=(1-cosx)/(1+cosx)`
`(csc^2x-2cotxcscx+cot^2x)=(1-cosx)/(1+cosx)`
`(1/sin^2x-(2cosx)/sin^2x+cos^2x/sin^2x)=(1-cosx)/(1+cosx)`

`((1-2cosx+cos^2x)/sin^2x)=(1-cosx)/(1+cosx)`

`(((1-cosx)(1-cosx))/(1-cos^2x))=(1-cosx)/(1+cosx)`

`((cancel(1-cosx)(1-cosx))/(cancel(1-cosx)(1+cosx)))=(1-cosx)/(1+cosx)`

`(1-cosx)/(1+cosx)=(1-cosx)/(1+cosx)`