document.write( "Question 1056873: Why is it important to work only 1 side of a trig identity?
\n" ); document.write( " ---
\n" ); document.write( " Can you show an example where it makes a difference?
\n" ); document.write( "--------------------
\n" ); document.write( "The question was about to scroll off, so I resubmitted it.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #671986 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "In a nutshell, it is always wrong to assume something that is not\r\n" );
document.write( "given. \r\n" );
document.write( "\r\n" );
document.write( "When we work with both sides of an equation, we are assuming that\r\n" );
document.write( "both sides are already equal.  But in the case of proving identities,\r\n" );
document.write( "we must begin WITHOUT assuming that both sides are equalfor \r\n" );
document.write( "\r\n" );
document.write( "It would be much better if a question mark were placed over the equal\r\n" );
document.write( "sign, i.e., that the symbol \r\n" );
document.write( "\r\n" );
document.write( "    ≟ \r\n" );
document.write( "\r\n" );
document.write( "were used instead of just an equal sign or identity symbol. That is, \r\n" );
document.write( "it would be better if textbooks and teachers would write this:\r\n" );
document.write( "\r\n" );
document.write( " ≟ \r\n" );
document.write( "\r\n" );
document.write( "instead of this  \r\n" );
document.write( "\r\n" );
document.write( " = \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The very fact that there is an equal or identity sign already \r\n" );
document.write( "there does cause students to get the false illusion that the two \r\n" );
document.write( "sides are already given equal, when that is not given, but rather\r\n" );
document.write( "is, instead, what is to be proved.\r\n" );
document.write( "\r\n" );
document.write( "Now your second question is a bit puzzling because it would always\r\n" );
document.write( "\"make a difference\" to do something mathematically incorrect instead\r\n" );
document.write( "of doing what is mathematically legal.  However if what is given is\r\n" );
document.write( "truly an identity which can proved, then even if we break the\r\n" );
document.write( "rules and go ahead and work with both sides, we will always get a\r\n" );
document.write( "result which is an equivalent equation.  So quite obviously we \r\n" );
document.write( "cannot start out with a true equation, work with both sides, and \r\n" );
document.write( "end up with a false equation!  This is why students who break the\r\n" );
document.write( "rules and work with both sides, end up with the same expression on\r\n" );
document.write( "both sides, and falsely think they have proved the identity, when \r\n" );
document.write( "they haven't at all.    \r\n" );
document.write( "\r\n" );
document.write( "Edwin

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

\n" );