document.write( "Question 1086313: If sin(x) + sin^2(x) = 1, then the value of cos^12(x) + 3cos^10(x) + 3cos^8(x) + cos^6(x) - 2 is ?
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Algebra.Com's Answer #700535 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Plots y = \"sin%28x%29+%2B+sin%5E2%28x%29\" (red) and y = 1 (green)\r
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document.write( "1.  \"sin%28x%29+%2B+sin%5E2%28x%29\" = 1  ====>  \"sin%5E2%28x%29+%2B+sin%28x%29+-1\" = 0  =====>\r\n" );
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document.write( "    solve this quadratic equation for sin(x) to get a single value for sin(x) = \"%28-1+%2B+sqrt%285%29%29%2F2\".\r\n" );
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document.write( "2.  \"sin%28x%29+%2B+sin%5E2%28x%29\" = 1  ====>  sin(x) = \"1+-+sin%5E2%28x%29\" = \"cos%5E2%28x%29\".\r\n" );
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document.write( "    Thus \"cos%5E2%28x%29\" = \"%28-1+%2B+sqrt%285%29%29%2F2\".\r\n" );
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document.write( "3.  \"cos%5E12%28x%29+%2B+3cos%5E10%28x%29+%2B+3cos%5E8%28x%29+%2B+cos%5E6%28x%29\" = \"cos%5E6%28x%29%2A%28cos%5E6%28x%29+%2B+3%2Acos%5E4%28x%29+%2B+3%2Acos%5E2%28x%29+%2B+1%29\" = \"cos%5E6%28x%29%2A%28cos%5E2%28x%29%2B1%29%5E3\" = \r\n" );
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document.write( "    = \"%28%28-1+%2B+sqrt%285%29%29%2F2%29%5E3\" . \"%28%281+%2B+sqrt%285%29%29%2F2%29%5E3\" = \"%28%285-1%29%2F4%29%5E3\" = 1.\r\n" );
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document.write( "4.  ====>  \"cos%5E12%28x%29+%2B+3cos%5E10%28x%29+%2B+3cos%5E8%28x%29+%2B+cos%5E6%28x%29+-+2\" = 1 - 2 = -1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. -1.\r
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