document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183638>Question 1183638</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove each of the following trigonometric identities.\r<BR>\n" );
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document.write( "1) sin x sin 2x + cox x cos 2x = cos x \r<BR>\n" );
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document.write( "2) cot x = sin x sin (pi/2 - x) + cos^2 x cot x \r<BR>\n" );
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document.write( "3) 2 csc 2x = sec x csc x  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814037 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814037\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Prove each of the following trigonometric identities.\r<BR>\n" );
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document.write( "1) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin+%28x%29+sin+%282x%29+%2B+cos+%28x+%29cos+%282x%29+=+cos%28+x%29 ALIGN=MIDDLE  ALT=\"sin+%28x%29+sin+%282x%29+%2B+cos+%28x+%29cos+%282x%29+=+cos%28+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "manipulate left side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin+%28x%29+%2Asin+%282x%29+%2B+cos+%28x+%29%2Acos+%282x%29 ALIGN=MIDDLE  ALT=\"sin+%28x%29+%2Asin+%282x%29+%2B+cos+%28x+%29%2Acos+%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin+%282x%29+=2sin%28x%29%2A+cos%28x%29+and+cos+%282x%29=cos%5E2%28x%29+-+sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"sin+%282x%29+=2sin%28x%29%2A+cos%28x%29+and+cos+%282x%29=cos%5E2%28x%29+-+sin%5E2%28x%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=sin+%28x%29+%2A2sin%28x%29%2A+cos%28x%29%2Bcos+%28x+%29%2A+%28cos%5E2%28x%29+-+sin%5E2%28x%29%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=2sin%5E2%28x%29%2A+cos%28x%29%2Bcos+%28x+%29%2A+%28cos%5E2%28x%29+-+sin%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"2sin%5E2%28x%29%2A+cos%28x%29%2Bcos+%28x+%29%2A+%28cos%5E2%28x%29+-+sin%5E2%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >...factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%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=cos%28x%29%282sin%5E2%28x%29%2B+%28cos%5E2%28x%29+-+sin%5E2%28x%29%29%29 ALIGN=MIDDLE  ALT=\"cos%28x%29%282sin%5E2%28x%29%2B+%28cos%5E2%28x%29+-+sin%5E2%28x%29%29%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=+cos%28x%29%282sin%5E2%28x%29%2B+cos%5E2%28x%29+-+sin%5E2%28x%29%29 ALIGN=MIDDLE  ALT=\"+cos%28x%29%282sin%5E2%28x%29%2B+cos%5E2%28x%29+-+sin%5E2%28x%29%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=cos%28x%29%28sin%5E2%28x%29%2B+cos%5E2%28x%29+%29 ALIGN=MIDDLE  ALT=\"cos%28x%29%28sin%5E2%28x%29%2B+cos%5E2%28x%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >............use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28x%29%2B+cos%5E2%28x%29=1 ALIGN=MIDDLE  ALT=\"sin%5E2%28x%29%2B+cos%5E2%28x%29=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=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "2) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot+%28x%29+=+sin+%28x+%29sin+%28pi%2F2+-+x%29+%2B+cos%5E2%28+x%29+cot%28+x%29 ALIGN=MIDDLE  ALT=\"cot+%28x%29+=+sin+%28x+%29sin+%28pi%2F2+-+x%29+%2B+cos%5E2%28+x%29+cot%28+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "manipulate right side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin+%28x+%29sin+%28pi%2F2+-+x%29+%2B+cos%5E2%28+x%29+cot%28+x%29 ALIGN=MIDDLE  ALT=\"sin+%28x+%29sin+%28pi%2F2+-+x%29+%2B+cos%5E2%28+x%29+cot%28+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F2-x%29=sin%28pi%2F2%29%2Acos%28x%29-cos%28pi%2F2%29%2Asin%28x%29=+cos%28x%29=1%2Acos%28x%29+-+0%2Asin%28x%29=cos%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=sin+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+cot%28+x%29 ALIGN=MIDDLE  ALT=\"sin+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+cot%28+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.... use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%28+x%29=cos%28x%29%2Fsin+%28x+%29 ALIGN=MIDDLE  ALT=\"cot%28+x%29=cos%28x%29%2Fsin+%28x+%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=sin+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+%28cos%28x%29%2Fsin+%28x+%29%29 ALIGN=MIDDLE  ALT=\"sin+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+%28cos%28x%29%2Fsin+%28x+%29%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=%28sin%5E2+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+%2Acos%28x%29%29%2Fsin+%28x+%29%29 ALIGN=MIDDLE  ALT=\"%28sin%5E2+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+%2Acos%28x%29%29%2Fsin+%28x+%29%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%28sin%5E2+%28x+%29%2B+cos%5E2%28+x%29%29+%2Acos%28x%29%29%2Fsin+%28x+%29%29 ALIGN=MIDDLE  ALT=\"%28%28sin%5E2+%28x+%29%2B+cos%5E2%28+x%29%29+%2Acos%28x%29%29%2Fsin+%28x+%29%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=%281+%2Acos%28x%29%29%2Fsin+%28x+%29%29 ALIGN=MIDDLE  ALT=\"%281+%2Acos%28x%29%29%2Fsin+%28x+%29%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=cos%28x%29%2Fsin+%28x+%29 ALIGN=MIDDLE  ALT=\"cos%28x%29%2Fsin+%28x+%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=cot%28+x%29 ALIGN=MIDDLE  ALT=\"cot%28+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "3) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2csc+%282x%29+=+sec+%28x%29+%2Acsc%28+x%29 ALIGN=MIDDLE  ALT=\"2csc+%282x%29+=+sec+%28x%29+%2Acsc%28+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "manipulate left side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2csc+%282x%29 ALIGN=MIDDLE  ALT=\"2csc+%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..................use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc+%282x%29+=+%281%2F2+%29csc%28x%29+sec%28x%29 ALIGN=MIDDLE  ALT=\"csc+%282x%29+=+%281%2F2+%29csc%28x%29+sec%28x%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=2%281%2F2%29csc%28x%29+%2Asec%28x%29%29 ALIGN=MIDDLE  ALT=\"2%281%2F2%29csc%28x%29+%2Asec%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .....simplify\r<BR>\n" );
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document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%28x%29+%2Asec%28x%29%29 ALIGN=MIDDLE  ALT=\"csc%28x%29+%2Asec%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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