Prove the trig identity?

(1+cosx)/(1-cosx) = tan^2x/(sec^2x-1)^2

1 Answer
Jan 14, 2018

Well, let's start from the left...

(1 + cosx)/(1 - cosx)

Multiply through by (1//cosx)/(1//cosx):

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

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

Use the fact that secx = 1/cosx, then multiply the top and bottom by secx - 1:

= (secx + 1)/(secx - 1)cdot (secx - 1)/(secx - 1)

= (sec^2x - 1)/(secx - 1)^2

Lastly, use the identity that 1 + tan^2x = sec^2x:

= (tan^2x)/(secx - 1)^2

color(red)(ne (tan^2x)/(sec^2x - 1)^2)

And we have shown that the identity is incorrect.