document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=90171>Question 90171</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I've been given this problem and I keep getting stuck half was down it and going back on myself, wondered i f anyone could help out with it.\r<BR>\n" );
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document.write( "Given that \r<BR>\n" );
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document.write( "cos x = (e^jx + e^-jx)/2    and\r<BR>\n" );
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document.write( "sin x = (e^jx - e^-jx)/2j\r<BR>\n" );
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document.write( "and using only this information, prove that \r<BR>\n" );
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document.write( "cos^2 x - sin^2 x = 1 - 2sin^2 x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #65445 by </B><A HREF=/tutors/aboutme.mpl?userid=kev82><B>kev82(151)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kev82><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=65445\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi<br />\r<BR>\n" );
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document.write( "First thing to do is to tidy up the thing we're trying to prove. Reason being, we're going to have to a bit of messy algebra, and the less we have to do, the better. I'm going to add sin^2(x) to both sides. This leaves the thing we have to prove at:<br />\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\cos^2(x) + \sin^2(x)=1\"><br />\r<BR>\n" );
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document.write( "Well, from here, all we do is substitute the definitions of sine and cosine in, then see what happens. To make it a bit simpler, I'm going to let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p=e^{jx}\;q=e^{-jx}\"> Using this, it's easy to see that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?pq=1\">.<br />\r<BR>\n" );
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document.write( "So using my new definitions, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sin(x) = \frac{p-q}{2j}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\cos(x) =\frac{p+q}{2}\"><br />\r<BR>\n" );
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document.write( "Substitute these into the thing we are trying to prove, and you should get:<br />\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\frac{(p+q)^2}{4} + \frac{(p-q)^2}{-4} = 1\"><br />\r<BR>\n" );
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document.write( "Multiply the whole lot by 4(we don't like denominators when we don't need them)<br />\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?(p+q)^2 - (p-q)^2 = 4\"><br />\r<BR>\n" );
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document.write( "That just leaves exapnding the brackets, and tidying up<br />\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?p^2 + 2pq + q^2 - (p^2 -2pq +q^2) = 4\"><br />\r<BR>\n" );
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document.write( "(Remember pq=1)<br />\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?(p^2-p^2) + (q^2-q^2) + (2pq - -2pq) = 4\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?4pq=4\">, so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?4=4\">. Which is always true, so what we started with must be true - proved.<br />\r<BR>\n" );
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document.write( "Now try a different one yourself, see if only using your definitions of sine and cosine you can prove that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sin(2x)=2\sin(x)\cos(x)\"><br />\r<BR>\n" );
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document.write( "Kev </SPAN>\n" );
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