Prove the trig identity?

Question

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

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Answer 1 · 2018-01-14T05:28:38

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.