document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=240783>Question 240783</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help prove the identity\r<BR>\n" );
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document.write( "cos(u-v)/cos(u)sin(v)= tan(u)+cot(v)\r<BR>\n" );
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document.write( "thank you!!!<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #176391 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=176391\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28u-v%29%2F%28cos%28u%29sin%28v%29%29=+tan%28u%29%2Bcot%28v%29 ALIGN=MIDDLE  ALT=\"cos%28u-v%29%2F%28cos%28u%29sin%28v%29%29=+tan%28u%29%2Bcot%28v%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since there is no (u-v) on the right side, we will use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A-B%29+=+cos%28A%29cos%28B%29+%2B+sin%28A%29sin%28B%29 ALIGN=MIDDLE  ALT=\"cos%28A-B%29+=+cos%28A%29cos%28B%29+%2B+sin%28A%29sin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the cos(u-v) giving us:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28u%29cos%28v%29+%2B+sin%28u%29sin%28v%29%29%2F%28cos%28u%29sin%28v%29%29=+tan%28u%29%2Bcot%28v%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since there are two terms on the right side, we will split the fraction on the left into two terms. (Think of it as \"unadding\".):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28u%29cos%28v%29%29%2F%28cos%28u%29sin%28v%29%29+%2B+%28sin%28u%29sin%28v%29%29%2F%28cos%28u%29sin%28v%29%29=+tan%28u%29%2Bcot%28v%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As you can see, we can do some canceling in each fraction:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cross%28cos%28u%29%29cos%28v%29%29%2F%28cross%28cos%28u%29%29sin%28v%29%29+%2B+%28sin%28u%29cross%28sin%28v%29%29%29%2F%28cos%28u%29cross%28sin%28v%29%29%29=+tan%28u%29%2Bcot%28v%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "leaving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28v%29%2Fsin%28v%29+%2B+sin%28u%29%2Fcos%28u%29=+tan%28u%29%2Bcot%28v%29 ALIGN=MIDDLE  ALT=\"cos%28v%29%2Fsin%28v%29+%2B+sin%28u%29%2Fcos%28u%29=+tan%28u%29%2Bcot%28v%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since cos(v)/sin(v) = cot(v) and sin(u)/cos(u) = tan(u) we can substitute and get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%28v%29+%2B+tan%28u%29+=+tan%28u%29%2Bcot%28v%29 ALIGN=MIDDLE  ALT=\"cot%28v%29+%2B+tan%28u%29+=+tan%28u%29%2Bcot%28v%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And, using the Commutative Property of Addition, we can change the order on the left to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28u%29%2B+cot%28v%29+=+tan%28u%29%2Bcot%28v%29 ALIGN=MIDDLE  ALT=\"tan%28u%29%2B+cot%28v%29+=+tan%28u%29%2Bcot%28v%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And we are done. </SPAN>\n" );
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