document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=883546>Question 883546</A>: <I><SPAN STYLE=\"word-break: break-all;\"> show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29sin%282x%29%2Bsin%283x%29sin%286x%29%29%2F%28sin%28x%29cos%282x%29%2Bsin%283x%29cos%286x%29%29=tan%285x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #533741 by </B><A HREF=/tutors/aboutme.mpl?userid=cigany29><B>cigany29(23)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=cigany29><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/cgi-bin/show-face.mpl?user=cigany29><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=533741\">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=%28sin%28x%29sin%282x%29%2Bsin%283x%29sin%286x%29%29%2F%28sin%28x%29cos%282x%29%2Bsin%283x%29cos%286x%29%29=tan%285x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Let's break it down first by applying our product-to-sum formulae!<BR>\n" );
document.write( "° <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29sin%282x%29%29 ALIGN=MIDDLE  ALT=\"%28sin%28x%29sin%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x-2x%29-cos%28x%2B2x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28cos%28x-2x%29-cos%28x%2B2x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28-x%29-cos%283x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28cos%28-x%29-cos%283x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "° <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%283x%29sin%286x%29%29 ALIGN=MIDDLE  ALT=\"%28sin%283x%29sin%286x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%283x-6x%29-cos%283x%2B6x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28cos%283x-6x%29-cos%283x%2B6x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28-3x%29-cos%289x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28cos%28-3x%29-cos%289x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "°<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29cos%282x%29%29 ALIGN=MIDDLE  ALT=\"%28sin%28x%29cos%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%2B2x%29%2Bsin%28x-2x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28sin%28x%2B2x%29%2Bsin%28x-2x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%283x%29%2Bsin%28-x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28sin%283x%29%2Bsin%28-x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "°<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%283x%29cos%286x%29%29 ALIGN=MIDDLE  ALT=\"%28sin%283x%29cos%286x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%283x%2B6x%29%2Bsin%283x-6x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28sin%283x%2B6x%29%2Bsin%283x-6x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%289x%29%2Bsin%28-3x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28sin%289x%29%2Bsin%28-3x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Remember:<BR>\n" );
document.write( "* cos(-x)=cos(x)<BR>\n" );
document.write( "*sin(-x)=-sin(x)\r<BR>\n" );
document.write( "\n" );
document.write( "SO:\r<BR>\n" );
document.write( "\n" );
document.write( "°<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28-x%29-cos%283x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28cos%28-x%29-cos%283x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29-cos%283x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28cos%28x%29-cos%283x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "°<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28-3x%29-cos%289x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28cos%28-3x%29-cos%289x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%283x%29-cos%289x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28cos%283x%29-cos%289x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "°<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%283x%29%2Bsin%28-x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28sin%283x%29%2Bsin%28-x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%283x%29-sin%28x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28sin%283x%29-sin%28x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "°<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%289x%29%2Bsin%28-3x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28sin%289x%29%2Bsin%28-3x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%289x%29-sin%283x%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28sin%289x%29-sin%283x%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now lets put it together!\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28cos%28x%29-cos%283x%29%2Bcos%283x%29-cos%289x%29%29%2F2%29%2F%28%28%28sin%283x%29-sin%28x%29%29%2B%28sin%289x%29-sin%283x%29%29%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "That looks kinda messy, so lets simplify that again and while we're at it, we'll get rid of one division by multiplying by the reciprocal!\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28cos%28x%29-cos%289x%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"%28%28cos%28x%29-cos%289x%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F%28sin%289x%29-sin%28x%29%29%29 ALIGN=MIDDLE  ALT=\"%282%2F%28sin%289x%29-sin%28x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Cancel the 2's and we're left with:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29-cos%289x%29%29%2F%28sin%289x%29-sin%28x%29%29%29 ALIGN=MIDDLE  ALT=\"%28cos%28x%29-cos%289x%29%29%2F%28sin%289x%29-sin%28x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now we're going to apply the sum-to-product formulae!\r<BR>\n" );
document.write( "\n" );
document.write( "°<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29-cos%289x%29%29 ALIGN=MIDDLE  ALT=\"%28cos%28x%29-cos%289x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2sin%28%28x%2B9x%29%2F2%29%2Asin%28%28x-9x%29%2F2%29%29 ALIGN=MIDDLE  ALT=\"%28-2sin%28%28x%2B9x%29%2F2%29%2Asin%28%28x-9x%29%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2sin%285x%29%2Asin%28-4x%29%29 ALIGN=MIDDLE  ALT=\"%28-2sin%285x%29%2Asin%28-4x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Again, sin(-x)=-sin(x) so it's = to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2sin%285x%29%2A%28-sin%284x%29%29%29 ALIGN=MIDDLE  ALT=\"%28-2sin%285x%29%2A%28-sin%284x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "°<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%289x%29-sin%28x%29%29 ALIGN=MIDDLE  ALT=\"%28sin%289x%29-sin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282cos%28%289x%2Bx%29%2F2%29%2Asin%28%289x-x%29%2F2%29%29 ALIGN=MIDDLE  ALT=\"%282cos%28%289x%2Bx%29%2F2%29%2Asin%28%289x-x%29%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282cos%285x%29%2Asin%284x%29%29 ALIGN=MIDDLE  ALT=\"%282cos%285x%29%2Asin%284x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now we'll put the whole thing together:<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2sin%285x%29%2A%28-sin%284x%29%29%29%2F%282cos%285x%29%2Asin%284x%29%29 ALIGN=MIDDLE  ALT=\"%28-2sin%285x%29%2A%28-sin%284x%29%29%29%2F%282cos%285x%29%2Asin%284x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%285x%29 ALIGN=MIDDLE  ALT=\"tan%285x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:\r<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%285x%29%29%2F%28cos%285x%29%29 ALIGN=MIDDLE  ALT=\"%28sin%285x%29%29%2F%28cos%285x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%285x%29 ALIGN=MIDDLE  ALT=\"tan%285x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and at last we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%285x%29=tan%285x%29 ALIGN=MIDDLE  ALT=\"tan%285x%29=tan%285x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );