Question

Prove it: `sqrt((1-cosx)/(1+cosx))+sqrt((1+cosx)/(1-cosx))=2/abs(sinx)`?

`sqrt((1-cosx)/(1+cosx))+sqrt((1+cosx)/(1-cosx))=2/abs(sinx)`

Answer 1

Proof below
using conjugates and trigonometric version of Pythagorean Theorem.

Explanation:

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

`color(white)("XXX")=sqrt(1-cosx)/sqrt(1+cosx)`

`color(white)("XXX")=sqrt((1-cosx))/sqrt(1+cosx) * sqrt(1-cosx)/sqrt(1-cosx)`

`color(white)("XXX")=(1-cosx)/sqrt(1-cos^2x)`

Part 2
Similarly
`sqrt((1+cosx)/(1-cosx)`

`color(white)("XXX")=(1+cosx)/sqrt(1-cos^2x)`

Part 3: Combining the terms
`sqrt((1-cosx)/(1+cosx))+sqrt((1+cosx)/(1-cosx)`

`color(white)("XXX")=(1-cosx)/sqrt(1-cos^2x)+(1+cosx)/sqrt(1-cos^2x)`

`color(white)("XXX")=2/sqrt(1-cos^2x)`

`color(white)("XXXXXX")`and since `sin^2x+cos^2x=1` (based on the Pythagorean Theorem)
`color(white)("XXXXXXXXX")sin^2x=1-cos^2x`

`color(white)("XXXXXXXXX")sqrt(1-cos^2x)=abs(sinx)`

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