How do you verify the Identity: $(1+cos x+ sinx)/(1+cos x-sin x)=sec x+tan x$ ?

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How do you verify the Identity: $(1+cos x+ sinx)/(1+cos x-sin x)=sec x+tan x$ ?

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Answer 1 · 2018-03-26T23:00:55

$(1+cosx+ sinx) / (1+cosx-sinx)=secx+tanx | * (1+cosx-sinx)$
$1+cosx+ sinx=(1/cosx+sinx/cosx)* (1+cosx-sinx)$
$=(1+cosx-sinx)/cosx+(1+cosx-sinx)*sinx/cosx$

$(1+cosx-sinx)/cosx=color(blue)(secx+1-tanx)$
$(1+cosx-sinx)*sinx/cosx=color(red)(tanx+sinx-sinxtanx)$

$1+cosx+ sinx=color(blue)(secx+1cancel(-tanx))+color(red)(cancel(tanx)+sinx-sinxtanx)$
$1+cosx+ sinx=secx+1+sinx-sinxtanx|-1-sinx$
$cosx=secx-sinxtanx|*cosx$
$cos^2x=1-sin^2x|+sin^2x$
$cos^2x+sin^2x=1$

Answer 2 · 2018-03-27T01:20:24

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

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

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

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

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

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

$=(1+sinx)/cosx$

$=1/cosx+sinx/cosx$

$=secx+tanx=RHS$

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How do you verify the Identity: $(1+cos x+ sinx)/(1+cos x-sin x)=sec x+tan x$ ?

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