document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=511681>Question 511681</A>: <I><SPAN STYLE=\"word-break: break-all;\"> where x represents the theta<BR>\n" );
document.write( "prove that sec x(1-sin x)(sec x+tan x) = 1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #342361 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=342361\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In this proof you can make use of three identities as follows:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29%2Asec%28theta%29+=+1 ALIGN=MIDDLE  ALT=\"cos%28theta%29%2Asec%28theta%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%28theta%29+=+1%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"sec%28theta%29+=+1%2Fcos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29+=+sin%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"tan%28theta%29+=+sin%28theta%29%2Fcos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2B+cos%5E2%28theta%29+=+1 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29%2B+cos%5E2%28theta%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28theta%29+=+1+-+sin%5E2%28theta%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28theta%29+=+1+-+sin%5E2%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Start with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sec%28theta%29%29%281-sin%28theta%29%29%28sec%28theta%29%2Btan%28theta%29%29+=+1 ALIGN=MIDDLE  ALT=\"%28sec%28theta%29%29%281-sin%28theta%29%29%28sec%28theta%29%2Btan%28theta%29%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "On the left side replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%28theta%29 ALIGN=MIDDLE  ALT=\"sec%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by its equivalent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"1%2Fcos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in two places to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fcos%28theta%29%29%281-sin%28theta%29%29%281%2Fcos%28theta%29%2B+tan%28theta%29%29=1 ALIGN=MIDDLE  ALT=\"%281%2Fcos%28theta%29%29%281-sin%28theta%29%29%281%2Fcos%28theta%29%2B+tan%28theta%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29 ALIGN=MIDDLE  ALT=\"tan%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by its equivalent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29%2Fcos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fcos%28theta%29%29%281-sin%28theta%29%29%281%2Fcos%28theta%29%2B+sin%28theta%29%2Fcos%28theta%29%29=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply the first two terms in parentheses and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281-sin%28theta%29%29%2Fcos%28theta%29%29%281%2Fcos%28theta%29%2B+sin%28theta%29%2Fcos%28theta%29%29=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In the second set of parentheses put everything over the common denominator and this makes the equation become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281-sin%28theta%29%29%2Fcos%28theta%29%29%28%281%2Bsin%28theta%29%29%2Fcos%28theta%29%29=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply the two numerators together. Also multiply the two denominators together:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281-sin%5E2%28theta%29%29%2Fcos%5E2%28theta%29%29=1 ALIGN=MIDDLE  ALT=\"%28%281-sin%5E2%28theta%29%29%2Fcos%5E2%28theta%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now replace the numerator by its equivalent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28theta%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the equation reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%5E2%28theta%29%2Fcos%5E2%28theta%29%29=1 ALIGN=MIDDLE  ALT=\"%28cos%5E2%28theta%29%2Fcos%5E2%28theta%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Dividing out the left side further reduces the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+1 ALIGN=MIDDLE  ALT=\"1+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And since the left side has been reduced until it equaled the right side, the proof has been validated. QED (thus it is demonstrated).<BR>\n" );
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document.write( "Hope this helps to familiarize you with some of the trig identities and how to use them. </SPAN>\n" );
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