document.write( "Question 686506: Prove the identity $\"%28cos+x+-+sin+x%29+%2F+%28cos+x+%2B+sin+x%29+=+sec+2+x+-+tan+2+x\"$ \n" ); document.write( "

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Prove the identity $\"%28cos+x+-+sin+x%29+%2F+%28cos+x+%2B+sin+x%29+=+sec+2+x+-+tan+2+x\"$
\n" ); document.write( "**
\n" ); document.write( "(cos x-sin x)/(cos x+sin x) = sec 2 x-tan 2 x
\n" ); document.write( "start with left side:
\n" ); document.write( "(cos x-sin x)/(cos x+sin x)
\n" ); document.write( "multiply top and bottom by (cos x-sin x), making the bottom a difference if 2 squares
\n" ); document.write( "(cos x-sin x)/(cos x+sin x)*(cos x-sin x)/(cos x-sin x)
\n" ); document.write( "=(cos x-sin x)(cos x-sin x)/(cos x+sin x)(cos x-sin x)
\n" ); document.write( "=(cos^2x-2sinxcosx+sin^2x)/(cos^2x-sin^2x)
\n" ); document.write( "=(1-2sinxcosx)/cos2x
\n" ); document.write( "=(1-sin2x)/cos2x
\n" ); document.write( "=(1/cos2x)-(sin2x/cos2x)
\n" ); document.write( "=sec2x-tan2x
\n" ); document.write( "verified: left side=right side \n" ); document.write( "

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