document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=693043>Question 693043</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm stumped on this Trigonometry Proof,\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cos%28x%29%2Fsec%28x%29%29+-+%28cot%28x%29%2Ftan%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "I've tried switching the denominators to their opposites, ie: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cot%28x%29%2Ftan%28x%29%29+ ALIGN=MIDDLE  ALT=\"+%28cot%28x%29%2Ftan%28x%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cot%5E2%28x%29%29+ ALIGN=MIDDLE  ALT=\"+%28cot%5E2%28x%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > . I've tried to substitute in some identities after that step too, but I can't get it.\r<BR>\n" );
document.write( "\n" );
document.write( " Thanks!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #427368 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=427368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cos%28x%29%2Fsec%28x%29%29+-+%28cot%28x%29%2Ftan%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You had a good idea with replacing the denominators. Replacing sec(x) with 1/cos(x) and tan(x) with 1/cot(x) we end up with:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28x%29+-+cot%5E2%28x%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29+-+cot%5E2%28x%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the right side is a product we want to be able to write the right side as a product, too. So we are looking to factor the left side. While the left side is a difference of squares, factoring it that way does not (as far as I can tell) get us closer to the end of the proof. Instead we should know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%5E2%28x%29+=+cos%5E2%28x%29%2Fsin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"cot%5E2%28x%29+=+cos%5E2%28x%29%2Fsin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%5E2%28x%29 ALIGN=MIDDLE  ALT=\"cot%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > this way we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28x%29+will+factor.+This+looks+very+promising+since+%7B%7B%7Bcos%5E2%28x%29+is+a+factor+on+the+right+side.%0D%0A%7B%7B%7Bcos%5E2%28x%29+-+cos%5E2%28x%29%2Fsin%5E2%28x%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If we're really cleaver we would factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-cos%5E2%28x%29 ALIGN=MIDDLE  ALT=\"-cos%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > since that is a factor of the right side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-cos%5E2%28x%29%28-1+%2B+1%2Fsin%5E2%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"-cos%5E2%28x%29%28-1+%2B+1%2Fsin%5E2%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can replace the fraction with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%5E2%28x%29 ALIGN=MIDDLE  ALT=\"csc%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-cos%5E2%28x%29%28-1+%2B+csc%5E2%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"-cos%5E2%28x%29%28-1+%2B+csc%5E2%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From one of the Pythagorean identities <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%5E2%28x%29+=+1+%2B+cot%5E2%28x%29 ALIGN=MIDDLE  ALT=\"csc%5E2%28x%29+=+1+%2B+cot%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-cos%5E2%28x%29%28-1+%2B+1+%2B+cot%5E2%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"-cos%5E2%28x%29%28-1+%2B+1+%2B+cot%5E2%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-cos%5E2%28x%29%28cot%5E2%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"-cos%5E2%28x%29%28cot%5E2%28x%29%29+=+%28-cos%5E2%28x%29%29%28cot%5E2%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And we are finished. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );