document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=138821>Question 138821</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove the identity: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%283x%29+=+3%2Asin%28x%29+%96+4%2A%28sin%28x%29%29%5E3 ALIGN=MIDDLE  ALT=\"sin%283x%29+=+3%2Asin%28x%29+%96+4%2A%28sin%28x%29%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (x is theta) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #101276 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=101276\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: I'm only manipulating the left side. I'm not touching the right side. I'm only showing it for comparison.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(3\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> 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 \sin\left(2\theta+\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Break up <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3\theta\"> to get <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\theta+\theta\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(2\theta\right)\cos\left(\theta\right)+\cos\left(2\theta\right)\sin\left(\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Expand using the sum-difference formulas\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2\sin\left(\theta\right)\cos\left(\theta\right)\cos\left(\theta\right)+\left(1-2\sin^2\left(\theta\right)\right)\sin\left(\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(2\theta\right)\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2\sin\left(\theta\right)\cos\left(\theta\right)\">. Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(2\theta\right)\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1-2\sin^2\left(\theta\right)\">.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\theta\right)\left(2\cos^2\left(\theta\right)+1-2\sin^2\left(\theta\right)\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Factor out <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\theta\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\theta\right)\left(2\cos^2\left(\theta\right)-2\sin^2\left(\theta\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Rearrange the terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\theta\right)\left(2\left(\cos^2\left(\theta\right)-\sin^2\left(\theta\right)\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Factor out 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\theta\right)\left(2\cos\left(2\theta\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos^2\left(\theta\right)-\sin^2\left(\theta\right)\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(2\theta\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\theta\right)\left(2\left(1-2\sin^2\left(\theta\right)\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(2\theta\right)\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1-2\sin^2\left(\theta\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\theta\right)\left(2-4\sin^2\left(\theta\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\theta\right)\left(3-4\sin^2\left(\theta\right)\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 3\sin\left(\theta\right)-4\sin^3\left(\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> Distribute\r<BR>\n" );
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document.write( "So we've just proven that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(3\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)\"> \r<BR>\n" );
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document.write( "is an identity.\r<BR>\n" );
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