document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=976230>Question 976230</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How to prove this following identity?<BR>\n" );
document.write( "(sin2x+cos2x)/(2cosx+sinx-2(cos^3x+sin^3x))=cosecx  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #598227 by </B><A HREF=/tutors/aboutme.mpl?userid=FrankM><B>FrankM(1040)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=FrankM><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=FrankM><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=598227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sin%282x%29%2Bcos%282x%29%29%2F%282cos%28x%29%2Bsin%28x%29-2%28cos%5E3+%28x%29%2Bsin%5E3+%28x%29%29%29=csc%28x%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First, the sum of cubes \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3%2Bb%5E3+=+%28a-b%29%28a%5E2-ab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"a%5E3%2Bb%5E3+=+%28a-b%29%28a%5E2-ab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This applies to the denominator \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cos%5E3%28x%29%2Bsin%5E3%28x%29%29+=+%28cos%28x%29%2Bsin%28x%29%29%28cos%5E2%28x%29-cos%28x%29sin%28x%29%2Bsin%5E2%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "but you see the sin squared and cos squared add to 1? \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cos%5E3%28x%29%2Bsin%5E3%28x%29%29+=+%28cos%28x%29%2Bsin%28x%29%29%281-cos%28x%29sin%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "simplify denominator a bit more to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sin%282x%29%2Bcos%282x%29%29%2F%28-sin%28x%29%2B2cos%5E2%28x%29sin%28x%29%2B2cos%28x%29sin%5E2%28x%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos2%28x%29=2cos%5E2%28x%29-1%29 ALIGN=MIDDLE  ALT=\"cos2%28x%29=2cos%5E2%28x%29-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and if we factor a sin out of the first two terms in denominator we get: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sin%282x%29%2Bcos%282x%29%29%2F%28sin%28x%29+%282cos%5E2%28x%29-1%29%2B2cos%28x%29sin%5E2%28x%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "and can make the substitution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sin%282x%29%2Bcos%282x%29%29%2F%28%28sin%28x%29+%28cos2%28x%29%29%29%2B2cos%28x%29sin%5E2%28x%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Next we have \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282x%29=2sin%28x%29cos%28x%29 ALIGN=MIDDLE  ALT=\"sin%282x%29=2sin%28x%29cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so let's factor sin out of the later part of denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sin%282x%29%2Bcos%282x%29%29%2F%28%28sin%28x%29+%28cos2%28x%29%29%29%2Bsin%28x%29+%282cos%28x%29sin%28x%29%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "and make the substitution\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sin%282x%29%2Bcos%282x%29%29%2F%28%28sin%28x%29+%28cos%282x%29%29%29%2Bsin%28x%29sin%282x%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "now, factor out sin in the denominator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sin%282x%29%2Bcos%282x%29%29%2F%28sin%28x%29+%28cos%282x%29%2Bsin%282x%29+%29%29+ ALIGN=MIDDLE  ALT=\"+%28sin%282x%29%2Bcos%282x%29%29%2F%28sin%28x%29+%28cos%282x%29%2Bsin%282x%29+%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "And after canceling numerator/denominator all we have left is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsin%28x%29 ALIGN=MIDDLE  ALT=\"1%2Fsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which is cosecant.\r<BR>\n" );
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document.write( "There may be a faster or more elegant method, but it's key to see that the sum of two cubes had to be factored to start the process.   </SPAN>\n" );
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