document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=132787>Question 132787</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help verifying this identity, factoring is confusing me.  Shouldn't I begin by factoring the top on the left side?  \r<BR>\n" );
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document.write( "cos 5x cos 3x + sin 5X sin 3x      CSC X<BR>\n" );
document.write( "-----------------------------  =  -------<BR>\n" );
document.write( "      sin 3x - sin X                 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #97032 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=97032\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: I'm <b>only</b> manipulating the left side. I'm showing the right just for comparison.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\cos \left(5x\right)\cos \left(3x\right)+\sin \left(5x\right)\sin \left(3x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\cos\left(5x-3x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Condense the numerator by using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-y%29=cos%28x%29cos%28y%29%2Bsin%28x%29sin%28y%29 ALIGN=MIDDLE  ALT=\"cos%28x-y%29=cos%28x%29cos%28y%29%2Bsin%28x%29sin%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\cos\left(2x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{\sin \left(3x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Use the identity <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos\left(2x\right)=1-2\sin^2\left(x\right)\"> to replace the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{\sin \left(2x+x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Expand 3x to 2x+x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{\sin\left(2x\right)\cos\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Expand <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282x%2Bx%29 ALIGN=MIDDLE  ALT=\"sin%282x%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > through this identity <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin\left(x+y\right)=\sin\left(x\right)\cos\left(y\right)+\cos\left(x\right)\sin\left(y\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)\cos\left(x\right)\cos\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282x%29 ALIGN=MIDDLE  ALT=\"sin%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the identity <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin\left(2x\right)=2\sin\left(x\right)\cos\left(x\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)\cos^2\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)\left(1-\sin^2\left(x\right)\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos^2\left(x\right)\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1-\sin^2\left(x\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)-2\sin^3\left(x\right)+\cos\left(2x\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)-2\sin^3\left(x\right)+\left(1-2\sin^2\left(x\right)\right)\sin\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos\left(2x\right)\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1-2\sin^2\left(x\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)-2\sin^3\left(x\right)+\sin\left(x\right)-2\sin^3\left(x\right)-\sin \left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)-4\sin^3\left(x\right)}=\frac{\csc\left(x\right)}{2}\">  Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1-2\sin^2\left(x\right)}{2\sin\left(x\right)\left(1-2\sin^2\left(x\right)\right)}=\frac{\csc\left(x\right)}{2}\"> Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\sin\left(x\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{2\sin\left(x\right)\left(1\right)}=\frac{\csc\left(x\right)}{2}\"> Cancel like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{2\sin\left(x\right)}=\frac{\csc\left(x\right)}{2}\"> Simplify \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\csc\left(x\right)}{2}=\frac{\csc\left(x\right)}{2}\"> replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsin%28x%29 ALIGN=MIDDLE  ALT=\"1%2Fsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%28x%29 ALIGN=MIDDLE  ALT=\"csc%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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