Question

Prove that `sqrt((1-cosx)/(1+cosx)) -= (1-cosx)/(|sinx|)` ?

Answer 1

Please see below.

Explanation:

.

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

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

Answer 2

We seek to prove that:

`sqrt((1-cosx)/(1+cosx)) -= (1-cosx)/(|sinx|)`

Consider the RHS:

`RHS = (1-cosx)/(|sinx|)`

`\ \ \ \ \ \ \ \ = sqrt( ((1-cosx)/(|sinx|))^2 )`

`\ \ \ \ \ \ \ \ = sqrt( (1-cosx)^2/(sin^2x) )`

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

`\ \ \ \ \ \ \ \ = sqrt( (1-cosx)^2/((1+cosx)(1-cosx) )`

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

`\ \ \ \ \ \ \ \ = LHS \ \ \ \` QED