document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=109302>Question 109302</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0136150624/algebrahomeworkh>Algebra and Trigonomometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Pove that<BR>\n" );
document.write( "1+sin(theta)divided by 1-Sin(theta)- 1-Sin(theta) Divided by 1+Sin(theta)=<BR>\n" );
document.write( "4Tan(theta)times Sec(theta)<BR>\n" );
document.write( "Show step by step how to change the left side of the equation to equal the right side of the equation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #79814 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=79814\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's only work on the left side of the equation. So we are working on:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281+%2B+sin%28x%29%29%2F%281-sin%28x%29%29%29+-+%28%281+-+sin%28x%29%29%2F%281%2Bsin%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "where x represents theta ... <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Multiply the first of these two terms by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%29%2F%281%2Bsin%28x%29%29 ALIGN=MIDDLE  ALT=\"%281%2Bsin%28x%29%29%2F%281%2Bsin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and multiply the second<BR>\n" );
document.write( "of these two terms by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-sin%28x%29%29%2F%281-sin%28x%29%29 ALIGN=MIDDLE  ALT=\"%281-sin%28x%29%29%2F%281-sin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Note that in each of these multipliers<BR>\n" );
document.write( "you are actually multiplying by 1 because the numerator equals the denominator. Therefore,<BR>\n" );
document.write( "in effect you are not changing the two terms.  We have now converted the problem to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281+%2B+sin%28x%29%29%2F%281-sin%28x%29%29%29%2A%28%281%2Bsin%28x%29%29%2F%281%2Bsin%28x%29%29%29+-+%28%281+-+sin%28x%29%29%2F%281%2Bsin%28x%29%29%29%2A%28%281-sin%28x%29%29%2F%281-sin%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice the products of the denominators ... in both terms the denominators multiply out<BR>\n" );
document.write( "to give 1 - sin^2(x). And if you recall the identity that sin^2(x) + cos^2(x) = 1, then<BR>\n" );
document.write( "you can see that cos^2(x) = 1 - sin^2(x). Therefore we can replace the denominators<BR>\n" );
document.write( "in the two terms with cos^2(x) which we split up into cos(x)*cos(x) and the problem then <BR>\n" );
document.write( "reduces to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281+%2B+sin%28x%29%29%2F%28cos%28x%29%29%29%2A%28%281%2Bsin%28x%29%29%2F%28cos%28x%29%29%29+-+%28%281+-+sin%28x%29%29%2Fcos%28x%29%29%2A%28%281-sin%28x%29%29%2F%28cos%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now let's multiply out the numerators in the two terms. The numerator of the first term<BR>\n" );
document.write( "is the product of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsin%28x%29%29%2A%281%2Bsin%28x%29%29 ALIGN=MIDDLE  ALT=\"%281%2Bsin%28x%29%29%2A%281%2Bsin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this multiplies out to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+2sin%28x%29+%2B+sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"1+%2B+2sin%28x%29+%2B+sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and the numerator of the second term is the product of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281+-+sin%28x%29%29%2A%281-sin%28x%29%29 ALIGN=MIDDLE  ALT=\"%281+-+sin%28x%29%29%2A%281-sin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "and this multiplies out to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-2sin%28x%29%2B+sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"1+-2sin%28x%29%2B+sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "These two results make the our expression become:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2B2sin%28x%29%2Bsin%5E2%28x%29%29%2F%28cos%28x%29%2Acos%28x%29%29%29-%28%281-2sin%28x%29%2B+sin%5E2%28x%29%29%2F%28cos%28x%29%2Acos%28x%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note that the denominators of the 2 terms are the same, so the numerators can be combined over <BR>\n" );
document.write( "the common denominator. This leads to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B2sin%28x%29%2Bsin%5E2%28x%29+-+%281+-2sin%28x%29%2B+sin%5E2%28x%29%29%29%2F%28cos%28x%29%2Acos%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "When you remove the parentheses around the second half of the numerator, the minus sign<BR>\n" );
document.write( "preceding those parentheses causes you to change the signs of all the terms inside the<BR>\n" );
document.write( "parentheses and you get:<BR>\n" );
document.write( ".\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B2sin%28x%29%2Bsin%5E2%28x%29+-1+%2B2sin%28x%29-+sin%5E2%28x%29%29%29%2F%28cos%28x%29%2Acos%28x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now in the numerator the 1 and the -1 cancel out and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"-sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "also cancel out. You are therefore left with:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282sin%28x%29+%2B+2+sin%28x%29%29%2F%28cos%28x%29%2Acos%28x%29%29 ALIGN=MIDDLE  ALT=\"%282sin%28x%29+%2B+2+sin%28x%29%29%2F%28cos%28x%29%2Acos%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And the terms in the numerator combine to give:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284sin%28x%29%29%2F%28cos%28x%29%2Acos%28x%29%29 ALIGN=MIDDLE  ALT=\"%284sin%28x%29%29%2F%28cos%28x%29%2Acos%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and we can separate this as follows:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A%28sin%28x%29%2Fcos%28x%29%29%2A%281%2Fcos%28x%29%29 ALIGN=MIDDLE  ALT=\"4%2A%28sin%28x%29%2Fcos%28x%29%29%2A%281%2Fcos%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "but by trig identities we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%2Fcos%28x%29+=+tan%28x%29 ALIGN=MIDDLE  ALT=\"sin%28x%29%2Fcos%28x%29+=+tan%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fcos%28x%29+=+sec%28x%29 ALIGN=MIDDLE  ALT=\"1%2Fcos%28x%29+=+sec%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Making these substitutions, we end up with:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A%28sin%28x%29%2Fcos%28x%29%29%2A%281%2Fcos%28x%29%29=+4%2Atan%28x%29%2Asec%28x%29 ALIGN=MIDDLE  ALT=\"4%2A%28sin%28x%29%2Fcos%28x%29%29%2A%281%2Fcos%28x%29%29=+4%2Atan%28x%29%2Asec%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And if you look back at the original problem, you find that this is the same as the right<BR>\n" );
document.write( "side of the identity you were to prove by showing that the left side could be converted to<BR>\n" );
document.write( "the right side ... which could be done as shown above. <BR>\n" );
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document.write( "Hope this is understandable and helps you to see your way through the problem.<BR>\n" );
document.write( ". </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );