How do you verify $(1-sinx)/(1+sinx)=sec^2x-2secxtanx+tan^2x$ ?

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Answer 1 · 2018-04-24T07:17:09

Please see the proof below

We need

$sin^2x+cos^2x=1$

$secx=1/cosx$

$tanx=sinx/cosx$

Therefore,

$LHS=(1-sinx)/(1+sinx)$

$=((1-sinx)(1-sinx))/((1+sinx)(1-sinx))$

$=(1-sinx)^2/(1-sin^2x)$

$=(1-2sinx+sin^2x)/(cos^2x)$

$=1/cos^2x-2sinx/cos^2x+sin^2x/cos^2x$

$=sec^2x-2secxtanx+tan^2x$

$=RHS$

$QED$

How do you verify (1-sinx)/(1+sinx)=sec^2x-2secxtanx+tan^2x (1-sinx)/(1+sinx)=sec^2x-2secxtanx+tan^2x?