document.write( "Question 1039841: If 2tanA + cot A =tanB then prove that 2 tan (B-A) =cot A \n" ); document.write( "
Algebra.Com's Answer #654750 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Quite often in mathematical proofs, we first do a \r\n" );
document.write( "\"motivation search\" for how to prove something \r\n" );
document.write( "by wrongly (yes wrongly!) and unjustifiably \r\n" );
document.write( "(yes unjustifiably!): \r\n" );
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document.write( "1. assuming what we want to prove is true, \r\n" );
document.write( "2. reaching something we know is true, and then \r\n" );
document.write( "3. seeing if we can start with what we found\r\n" );
document.write( "   to be true, and\r\n" );
document.write( "4. reverse the steps of our 'wrong and unjustifiable' \r\n" );
document.write( "   trick to create a \"correct and justifiable' proof. \r\n" );
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document.write( "So let's first do a 'motivation search' by 'unjustifiably'\r\n" );
document.write( "assuming what we want to prove. That is we assume this: \r\n" );
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document.write( "\"2tan%28B-A%29\"\"%22%22=%22%22\"\"cot%28A%29\"\r\n" );
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document.write( "Using well known identities:\r\n" );
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document.write( "\"2%28+%28tan%28B%29-tan%28A%29%29%2F%281%2Btan%28A%29tan%28B%29%29%29\"\"%22%22=%22%22\"\"1%2Ftan%28A%29\"\r\n" );
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document.write( "\"%282tan%28B%29-2tan%28A%29%29%2F%281%2Btan%28A%29tan%28B%29%29\"\"%22%22=%22%22\"\"1%2Ftan%28A%29\"\r\n" );
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document.write( "Cross-multiply:\r\n" );
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document.write( "\"tan%28A%29%282tan%28B%29%5E%22%22-2tan%28A%29%29\"\"%22%22=%22%22\"\"1%2Btan%28A%29tan%28B%29\"\r\n" );
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document.write( "\"2tan%28A%29tan%28B%29-2tan%5E2%28A%29\"\"%22%22=%22%22\"\"1%2Btan%28A%29tan%28B%29\"\r\n" );
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document.write( "Subtract tan(A)tan(B) from both sides:\r\n" );
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document.write( "\"tan%28A%29tan%28B%29-2tan%5E2%28A%29\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "Add 2tanē(A) to both sides:\r\n" );
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document.write( "(1)   \"tan%28A%29tan%28B%29\"\"%22%22=%22%22\"\"1%2B2tan%5E2%28A%29\"\r\n" );
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document.write( "Now let's use the given equation: \r\n" );
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document.write( "\"2tan%28A%29+%2B+cot%28A%29\"\"%22%22=%22%22\"\"tan%28B%29\"\r\n" );
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document.write( "Replacing the cot(A),\r\n" );
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document.write( "\"2tan%28A%29%2B1%2Ftan%28A%29\"\"%22%22=%22%22\"\"tan%28B%29\"              \r\n" );
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document.write( "and substitute for tan(B) in equation (1) above:\r\n" );
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document.write( " \"tan%28A%29%282tan%28A%29%2B1%2Ftan%28A%29%29\"\"%22%22=%22%22\"\"1%2B2tan%5E2%28A%29\"\r\n" );
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document.write( " \"2tan%5E2%28A%29%2B1\"\"%22%22=%22%22\"\"1%2B2tan%5E2%28A%29\"\r\n" );
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document.write( "Now we have reached an equation which we know is \r\n" );
document.write( "true.  But please realize that we haven't proved \r\n" );
document.write( "anything at all! We haven't even begun to prove \r\n" );
document.write( "anything! So now we start our proof by reversing \r\n" );
document.write( "our steps.\r\n" );
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document.write( "We know this is true:\r\n" );
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document.write( " \"2tan%5E2%28A%29%2B1\"\"%22%22=%22%22\"\"1%2B2tan%5E2%28A%29\"\r\n" );
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document.write( "On the left, break up tan2(A) as tan(A)tan(A)\r\n" );
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document.write( "\"2tan%28A%29tan%28A%29%2B1\"\"%22%22=%22%22\"\"1%2B2tan%5E2%28A%29\"\r\n" );
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document.write( "Factor out tan(A) on the left\r\n" );
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document.write( " \"tan%28A%29%282tan%28A%29%2B1%2Ftan%28A%29%29\"\"%22%22=%22%22\"\"1%2B2tan%5E2%28A%29\"\r\n" );
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document.write( "Replace \"1%2Ftan%28A%29\" by cot(A)\r\n" );
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document.write( " \"tan%28A%29%282tan%28A%29%5E%22%22%2Bcot%28A%29%29\"\"%22%22=%22%22\"\"1%2B2tan%5E2%28A%29\"\r\n" );
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document.write( "We are given that \"2tan%28A%29+%2B+cot%28A%29\"\"%22%22=%22%22\"\"tan%28B%29\", so\r\n" );
document.write( "we can replace what's in the parentheses by tan(B)\r\n" );
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document.write( " \"tan%28A%29tan%28B%29\"\"%22%22=%22%22\"\"1%2B2tan%5E2%28A%29\" \r\n" );
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document.write( "Subtract 2tanē(A) from both sides:\r\n" );
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document.write( " \"tan%28A%29tan%28B%29-2tan%5E2%28A%29\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "Add tan(A)tan(B) to both sides:\r\n" );
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document.write( " \"tan%28A%29tan%28B%29%2Btan%28A%29tan%28B%29-2tan%5E2%28A%29\"\"%22%22=%22%22\"\"1%2Btan%28A%29tan%28B%29\"\r\n" );
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document.write( " \"2tan%28A%29tan%28B%29-2tan%5E2%28A%29\"\"%22%22=%22%22\"\"1%2Btan%28A%29tan%28B%29\"\r\n" );
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document.write( "Factor out 2tan(A) on the left:\r\n" );
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document.write( " \"2tan%28A%29%28tan%28B%29%5E%22%22-tan%28A%29%29\"\"%22%22=%22%22\"\"1%2Btan%28A%29tan%28B%29\"\r\n" );
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document.write( "Divide both sides by 1+tan(A)tan(B)\r\n" );
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document.write( " \"2tan%28A%29%28tan%28B%29%5E%22%22-tan%28A%29%29%2F%281%2Btan%28A%29tan%28B%29%29\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "Divide both sides by tan(A)\r\n" );
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document.write( " \"2%28%28tan%28B%29%5E%22%22-tan%28A%29%29%2F%281%2Btan%28A%29tan%28B%29%29%29\"\"%22%22=%22%22\"\"1%2Ftan%28A%29\"\r\n" );
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document.write( "Replace the expression in parentheses by tan(B-A) on left,\r\n" );
document.write( "Replace \"1%2Ftan%28A%29\" by cot(A) on the right:\r\n" );
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document.write( " \"2tan%28B%29-A\"\"%22%22=%22%22\"\"cot%28A%29\"\r\n" );
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document.write( "Notice that we would never have guessed what true equation\r\n" );
document.write( "we should start with and what manipulations to do with it \r\n" );
document.write( "if we hadn't begun by 'wrongly' assuming what we wanted to \r\n" );
document.write( "prove.  This is a technique useful for many proofs.\r\n" );
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document.write( "Edwin
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