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

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Question 686506: Prove the identity %28cos+x+-+sin+x%29+%2F+%28cos+x+%2B+sin+x%29+=+sec+2+x+-+tan+2+x
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
Prove the identity %28cos+x+-+sin+x%29+%2F+%28cos+x+%2B+sin+x%29+=+sec+2+x+-+tan+2+x
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(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