SOLUTION: Prove the identity {{{(cos x - sin x) / (cos x + sin x) = sec 2 x

SOLUTION: Prove the identity {{{(cos x - sin x) / (cos x + sin x) = sec 2 x - tan 2 x}}}

Question 686506: Prove the identity
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
Prove the identity
**
(cos x-sin x)/(cos x+sin x) = sec 2 x-tan 2 x
start with left side:
(cos x-sin x)/(cos x+sin x)
multiply top and bottom by (cos x-sin x), making the bottom a difference if 2 squares
(cos x-sin x)/(cos x+sin x)*(cos x-sin x)/(cos x-sin x)
=(cos x-sin x)(cos x-sin x)/(cos x+sin x)(cos x-sin x)
=(cos^2x-2sinxcosx+sin^2x)/(cos^2x-sin^2x)
=(1-2sinxcosx)/cos2x
=(1-sin2x)/cos2x
=(1/cos2x)-(sin2x/cos2x)
=sec2x-tan2x
verified: left side=right side
RELATED QUESTIONS
Prove the following identity. (sin x * tan x + cos x)/ (cos x) = sec^2... (answered by robertb)

Prove the identity: {{{cos(x)/(1-sin(x)) + cos(x)/(1+sin(x)) =... (answered by DAWNPLAYA,Boreal)

Prove the identity:... (answered by math_tutor2020)

prove identity sin x tan x + cos x = sec... (answered by Edwin McCravy)

Prove the following trig identity: tan x *sin^2 x + sin x*cos x = tan... (answered by greenestamps)

Sin(x).tan(x)/cos(x) + 1 = 1/cos^2(x) Prove the... (answered by Alan3354)

please help me prove the identity... (answered by Timnewman,Alan3354)

Prove the identity: {{{tan(x) /( 1 + (tan(x))^2) = sin(x)cos(x)}}} (x is... (answered by jim_thompson5910)

prove the identity of (sin x)(cot x +cos x tan x)=cos x + sin^2... (answered by stanbon)