document.write( "Question 891903: cos3xcos5x-cos7xcos9x=0 \n" ); document.write( "
Algebra.Com's Answer #540154 by Edwin McCravy(20081)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"cos%283x%29cos%285x%29-cos%287x%29cos%289x%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "We use the product to sum formula on each term\r\n" );
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document.write( "\"cos%28alpha%29cos%28beta%29\"\"%22%22=%22%22\"\"expr%281%2F2%29%28cos%28alpha%2Bbeta%29%2Bcos%28alpha-beta%29%29\"\r\n" );
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document.write( "Swap the factors in each product so the difference will be positive:\r\n" );
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document.write( "\"cos%285x%29cos%283x%29-cos%289x%29cos%287x%29\"\"%22%22=%22%22\"\"%220%22\" \r\n" );
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document.write( "\"expr%281%2F2%29%28cos%285x%2B3x%29%2Bcos%285x-3x%29%29\"\"%22%22-%22%22\"\"expr%281%2F2%29%28cos%289x%2B7x%29%2Bcos%289x-7x%29%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "\"expr%281%2F2%29%28cos%288x%29%2Bcos%282x%29%29\"\"%22%22-%22%22\"\"expr%281%2F2%29%28cos%2816x%29%2Bcos%282x%29%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "\"%28cos%288x%29%2Bcos%282x%29%29-%28cos%2816x%29%2Bcos%282x%29%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "\"cos%288x%29%2Bcos%282x%29-cos%2816x%29-cos%282x%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "\"cos%288x%29%2Bcross%28cos%282x%29%29-cos%2816x%29-cross%28cos%282x%29%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "\"cos%288x%29-cos%2816x%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "Use the doubla angle identity \"cos%282theta%29\"\"%22%22=%22%22\"\"2cos%5E2%28theta%29-1\"\r\n" );
document.write( "with \"theta=8x\", \"2theta=16x\"\r\n" );
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document.write( "\"cos%288x%29-%282cos%5E2%288x%29-1%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "\"cos%288x%29-2cos%5E2%288x%29%2B1\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "\"-2cos%5E2%288x%29%2Bcos%288x%29%2B1\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "Multiply through by -1\r\n" );
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document.write( "\"2cos%5E2%288x%29-cos%288x%29-1\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "Factor the left side:\r\n" );
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document.write( "\"%282cos%288x%29%2B1%29%28cos%288x%29-1%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
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document.write( "Use zero-factor property:\r\n" );
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document.write( "\"2cos%288x%29%2B1=0\",     \"cos%288x%29-1=0\"\r\n" );
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document.write( "\"2cos%288x%29=-1\",     \"cos%288x%29=1\"\r\n" );
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document.write( "\"cos%288x%29=-1%2F2\",     \"8x=2n%2Api\"\r\n" );
document.write( "                    \r\n" );
document.write( "\"8x=+pi+%2B-+pi%2F3%2B2n%2Api\",     \"x=expr%28n%2F4%29%2Api\"       \r\n" );
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document.write( "\"8x+=+%281+%2B-+1%2F3+%2B+2n%29pi\"\r\n" );
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document.write( "\"8x=+%283%2F3+%2B-+1%2F3+%2B6n%2F3%29pi\"\r\n" );
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document.write( "\"8x+=+%28%283+%2B-+1%2B6n%29%2F3%29pi\"\r\n" );
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document.write( "\"8x+=+%28%286n+%2B3+%2B-+1%29%2F3%29pi\"\r\n" );
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document.write( "\"x+=+%28%286n+%2B3+%2B-+1%29%2F24%29pi\",     \"x=expr%28n%2F4%29%2Api\"\r\n" );
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document.write( "Edwin
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