document.write( "Question 1191899: if cot(tan(x)) =tan(cot(x)), then sin(2x) =..... \n" ); document.write( "
Algebra.Com's Answer #823755 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"cot%28tan%28x%29%5E%22%22%29\"\"%22%22=%22%22\"\"tan%28cot%28x%29%5E%22%22%29\"\r\n" );
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document.write( "\"cos%28tan%28x%29%5E%22%22%29%2Fsin%28tan%28x%29%5E%22%22%29\"\"%22%22=%22%22\"\"sin%28cot%28x%29%5E%22%22%29%2Fcos%28cot%28x%29%5E%22%22%29\"\r\n" );
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document.write( "Cross-multiply:\r\n" );
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document.write( "\"cos%28tan%28x%29%5E%22%22%29cos%28cot%28x%29%5E%22%22%29\"\"%22%22=%22%22\"\"sin%28cot%28x%29%5E%22%22%29sin%28tan%28x%29%5E%22%22%29\"\r\n" );
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document.write( "Get 0 on the right side:\r\n" );
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document.write( "\"%22%22=%22%22\"\"0\"\r\n" );
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document.write( "Use the identity \"cos%28alpha%2Bbeta%29\"\"%22%22=%22%22\"\"cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29\"\r\n" );
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document.write( "\"cos%28tan%28x%29%2Bcot%28x%29%5E%22%22%29\"\"%22%22=%22%22\"\"0\"\r\n" );
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document.write( "So tan(x)+cot(x) must equal to any odd integer multiple of \"pi%2F2\"\r\n" );
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document.write( "The general odd integer is 2n+1 for any integer n\r\n" );
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document.write( "\"tan%28x%29%2Bcot%28x%29\"\"%22%22=%22%22\"\"%282n%2B1%29%28pi%2F2%29\"\r\n" );
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document.write( "\"sin%28x%29%2Fcos%28x%29%2Bcos%28x%29%2Fsin%28x%29\"\"%22%22=%22%22\"\"%282n%2B1%29%28pi%2F2%29\"\r\n" );
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document.write( "\"%28sin%5E2%28x%29%2Bcos%5E2%28x%29%29%2F%28cos%28x%29sin%28x%29%29\"\"%22%22=%22%22\"\"%282n%2B1%29%28pi%2F2%29\"\r\n" );
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document.write( "\"1%2F%28cos%28x%29sin%28x%29%29\"\"%22%22=%22%22\"\"%282n%2B1%29%28pi%2F2%29\"\r\n" );
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document.write( "Take reciprocals of both sides:\r\n" );
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document.write( "\"cos%28x%29sin%28x%29\"\"%22%22=%22%22\"\"1%2F%28%282n%2B1%29%28pi%2F2%29%29\"\r\n" );
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document.write( "Multiply numerator and denominator on the right side by 2\r\n" );
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document.write( "\"cos%28x%29sin%28x%29\"\"%22%22=%22%22\"\"2%2F%28%282n%2B1%29pi%29\"\r\n" );
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document.write( "Multiply both sides by 2 and reverse the factors on the left side\r\n" );
document.write( "to make the left side look like the right side of the \r\n" );
document.write( "identity \"sin%282x%29\"\"%22%22=%22%22\"\"2sin%28x%29cos%28x%29\"\r\n" );
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document.write( "\"2sin%28x%29cos%28x%29\"\"%22%22=%22%22\"\"4%2F%28%282n%2B1%29pi%29\"\r\n" );
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document.write( "\"sin%282x%29\"\"%22%22=%22%22\"\"4%2F%28%282n%2B1%29pi%29\", where n is any integer.\r\n" );
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document.write( "Edwin
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