\r\n" );
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" );
document.write( "\r\n" );
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" );
document.write( "\r\n" );
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( "

\r\n" );
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document.write( "Using well known identities:\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "Cross-multiply:\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
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document.write( "Subtract tan(A)tan(B) from both sides:\r\n" );
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document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "Add 2tanē(A) to both sides:\r\n" );
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document.write( "(1) 

\r\n" );
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document.write( "Now let's use the given equation: \r\n" );
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document.write( "

\r\n" );
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document.write( "Replacing the cot(A),\r\n" );
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document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "and substitute for tan(B) in equation (1) above:\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( " 

\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( "-------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "We know this is true:\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( "On the left, break up tan2(A) as tan(A)tan(A)\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "Factor out tan(A) on the left\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( "Replace
by cot(A)\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
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document.write( "We are given that 

, so\r\n" );
document.write( "we can replace what's in the parentheses by tan(B)\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( "Subtract 2tanē(A) from both sides:\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( "Add tan(A)tan(B) to both sides:\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( " 

\r\n" );
document.write( "\r\n" );
document.write( "Factor out 2tan(A) on the left:\r\n" );
document.write( "\r\n" );
document.write( " 

\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( " 

\r\n" );
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document.write( "Divide both sides by tan(A)\r\n" );
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document.write( " 

\r\n" );
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document.write( "Replace the expression in parentheses by tan(B-A) on left,\r\n" );
document.write( "Replace
by cot(A) on the right:\r\n" );
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document.write( " 

\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" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "