document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152684>Question 1152684</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bicos%28x%29%29%2F%281%2Bsin%28x%29-icos%28x%29%29 ALIGN=MIDDLE  ALT=\"%281%2Bsin%28x%29%2Bicos%28x%29%29%2F%281%2Bsin%28x%29-icos%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = sin(x) + icos(x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #774758 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=774758\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The left hand side (LHS) is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2BB%29%2F%28A-B%29 ALIGN=MIDDLE  ALT=\"%28A%2BB%29%2F%28A-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+1%2Bsin%28x%29 ALIGN=MIDDLE  ALT=\"A+=+1%2Bsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B+=+i%2Acos%28x%29 ALIGN=MIDDLE  ALT=\"B+=+i%2Acos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Throughout this entire problem, I will not alter the right hand side (RHS). When it comes to proving identities, the method is to pick one side and transform it into the other. Whatever side you havent picked to transform will remain the same the entire time. \r<BR>\n" );
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document.write( "Multiply top and bottom of the LHS by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2BB%29%2F%28A%2BB%29 ALIGN=MIDDLE  ALT=\"%28A%2BB%29%2F%28A%2BB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that we transform the denominator into a real number\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%2F%281%2Bsin%28x%29-i%2Acos%28x%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Original equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%28A%2BB%29%29%2F%28%281%2Bsin%28x%29-i%2Acos%28x%29%29%28A%2BB%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply top and bottom by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2BB%29%2F%28A%2BB%29 ALIGN=MIDDLE  ALT=\"%28A%2BB%29%2F%28A%2BB%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/plot-formula.mpl?expression=%28%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%29%2F%28%281%2Bsin%28x%29-i%2Acos%28x%29%29%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in values of A and B.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2%29%2F%28%281%2Bsin%28x%29%29%5E2-%28i%2Acos%28x%29%29%5E2%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Numerator is a perfect square. Difference of squares rule in the denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2%29%2F%281%2B2sin%28x%29%2Bsin%5E2%28x%29%2Bcos%5E2%28x%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Use the idea that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2+=+-1 ALIGN=MIDDLE  ALT=\"i%5E2+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2%29%2F%281%2B2sin%28x%29%2B1%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Pythagorean trig identity: sin^2+cos^2 = 1 (denominator LHS)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2%29%2F%282%2B2sin%28x%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2%29%2F%282%281%2Bsin%28x%29%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Factor\r<BR>\n" );
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document.write( "Let's take a brief pause from that. We'll come back to it of course. The numerator of the LHS will get a bit ugly when we expand it out, so let's just focus on that for now.  \r<BR>\n" );
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document.write( "We have something in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2BB%29%5E2 ALIGN=MIDDLE  ALT=\"%28A%2BB%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+1%2Bsin%28x%29 ALIGN=MIDDLE  ALT=\"A+=+1%2Bsin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B+=+i%2Acos%28x%29 ALIGN=MIDDLE  ALT=\"B+=+i%2Acos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use the rule that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%2BB%29%5E2+=+A%5E2+%2B+2AB+%2B+B%5E2 ALIGN=MIDDLE  ALT=\"%28A%2BB%29%5E2+=+A%5E2+%2B+2AB+%2B+B%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So,<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+%281%2Bsin%28x%29%29%5E2+%2B+2%281%2Bsin%28x%29%29%2Ai%2Acos%28x%29+%2B+%28i%2Acos%28x%29%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the previously mentioned values of A and B\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+%281%2Bsin%28x%29%29%5E2+%2B+2i%2Acos%28x%29%2B2i%2Asin%28x%29cos%28x%29-cos%5E2%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Use i^2 = -1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+1%2B2sin%28x%29%2Bsin%5E2%28x%29+%2B+2i%2Acos%28x%29%2B2i%2Asin%28x%29cos%28x%29-cos%5E2%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Expand (1+sin(x))^2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+1%2B2sin%28x%29%2B1-cos%5E2%28x%29+%2B+2i%2Acos%28x%29%2B2i%2Asin%28x%29cos%28x%29-cos%5E2%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Use sin^2 = 1-cos^2 (variation of the pythagorean trig identity)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+2%2B2sin%28x%29+%2B+2i%2Acos%28x%29%2B2i%2Asin%28x%29cos%28x%29-2cos%5E2%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms (pair of '1's, also pair of -cos^2 terms)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+2%281%2Bsin%28x%29+%2B+i%2Acos%28x%29%2Bi%2Asin%28x%29cos%28x%29-cos%5E2%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+2%28sin%5E2%28x%29%2Bcos%5E2%28x%29%2Bsin%28x%29+%2B+i%2Acos%28x%29%2Bi%2Asin%28x%29cos%28x%29-cos%5E2%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Replace the term '1' with sin^2+cos^2, which helps with the canceling on the next step.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+2%28sin%5E2%28x%29%2Bsin%28x%29+%2B+i%2Acos%28x%29%2Bi%2Asin%28x%29cos%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. Cos^2 terms cancel.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+2%28sin%28x%29%28sin%28x%29%2B1%29+%2B+i%2Acos%28x%29%281%2Bsin%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Pair up terms, then factor by grouping\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+2%28sin%28x%29%2Bi%2Acos%28x%29%29%28sin%28x%29%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Complete the factor by grouping process\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2+=+2%28sin%28x%29%2B1%29%28sin%28x%29%2Bi%2Acos%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange terms. We'll use this later. Call this equation (1).\r<BR>\n" );
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document.write( "There is possibly a more efficient method to expanding out this expression, but I can't think of it right now.\r<BR>\n" );
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document.write( "Back to the main event<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Bsin%28x%29%2Bi%2Acos%28x%29%29%5E2%29%2F%282%281%2Bsin%28x%29%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%28sin%28x%29%2B1%29%28sin%28x%29%2Bi%2Acos%28x%29%29%29%2F%282%281%2Bsin%28x%29%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Replace the numerator on the LHS using equation (1)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28highlight%282%2A%28sin%28x%29%2B1%29%29%28sin%28x%29%2Bi%2Acos%28x%29%29%29%2F%28highlight%282%2A%28sin%28x%29%2B1%29%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > We have these pairs of terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cross%282%2A%28sin%28x%29%2B1%29%29%28sin%28x%29%2Bi%2Acos%28x%29%29%29%2F%28cross%282%2A%28sin%28x%29%2B1%29%29%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > that cancel out\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%2Bi%2Acos%28x%29+=+sin%28x%29+%2B+i%2Acos%28x%29 ALIGN=MIDDLE  ALT=\"sin%28x%29%2Bi%2Acos%28x%29+=+sin%28x%29+%2B+i%2Acos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > The LHS has been transformed into the RHS. The identity is confirmed.<BR>\n" );
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