document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=217335>Question 217335</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cotA+%2B+1%29%2F+%28cotA+-+1%29+ ALIGN=MIDDLE  ALT=\"+%28cotA+%2B+1%29%2F+%28cotA+-+1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Hence show that cot15° = 2 + sqrt3  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #163907 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=163907\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cotA+%2B+1%29%2F+%28cotA+-+1%29+ ALIGN=MIDDLE  ALT=\"+%28cotA+%2B+1%29%2F+%28cotA+-+1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There may be more clever ways of doing this but one approach to a lot of Trig. problems is to start by rewriting the expression in terms of sin and cos. Since cot(a) = cos(a)/sin(a):<BR>\n" );
document.write( "cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+cos%2845+-A%29%2Fsin%2845-A%29+ ALIGN=MIDDLE  ALT=\"+cos%2845+-A%29%2Fsin%2845-A%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can use the angle difference identitites: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-y%29+=+cos%28x%29%2Acos%28y%29+%2B+sin%28x%29%2Asin%28y%29 ALIGN=MIDDLE  ALT=\"cos%28x-y%29+=+cos%28x%29%2Acos%28y%29+%2B+sin%28x%29%2Asin%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x-y%29+=+sin%28x%29%2Acos%28y%29+-+cos%28x%29sin%28y%29 ALIGN=MIDDLE  ALT=\"sin%28x-y%29+=+sin%28x%29%2Acos%28y%29+-+cos%28x%29sin%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to rewrite the right side:<BR>\n" );
document.write( "cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%2845%29cos%28A%29+%2B+sin%2845%29sin%28A%29%29%2F%28sin%2845%29cos%28A%29-cos%2845%29sin%28A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2845%29+=+sin%2845%29+=+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"cos%2845%29+=+sin%2845%29+=+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28sqrt%282%29%2F2%29cos%28A%29+%2B+%28sqrt%282%29%2F2%29sin%28A%29%29%2F%28%28sqrt%282%29%2F2%29cos%28A%29-%28sqrt%282%29%2F2%29sin%28A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can reduce the fraction by factoring out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28sqrt%282%29%2F2%29%28cos%28A%29+%2B+sin%28A%29%29%29%2F%28%28sqrt%282%29%2F2%29%28cos%28A%29-sin%28A%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2F2%29 ALIGN=MIDDLE  ALT=\"%28sqrt%282%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s cancel leaving:<BR>\n" );
document.write( "cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28A%29+%2B+sin%28A%29%29%2F%28cos%28A%29-sin%28A%29%29 ALIGN=MIDDLE  ALT=\"%28cos%28A%29+%2B+sin%28A%29%29%2F%28cos%28A%29-sin%28A%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Looking at what we have and at where we want to be we can see that we're very close. All we need is sin(A) as a denominator of each term. So we multiply the top and bottom by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsin%28A%29 ALIGN=MIDDLE  ALT=\"1%2Fsin%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2Fsin%28A%29%29%2F%281%2Fsin%28A%29%29%29%28%28cos%28A%29+%2B+sin%28A%29%29%2F%28cos%28A%29-sin%28A%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the Distributive Property we get:<BR>\n" );
document.write( "cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28A%29%2Fsin%28A%29+%2B+sin%28A%29%2Fsin%28A%29%29%2F%28cos%28A%29%2Fsin%28A%29+-+sin%28A%29%2Fsin%28A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to<BR>\n" );
document.write( "cot(45°-A) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cot%28A%29+%2B+1%29%2F+%28cot%28A%29+-+1%29+ ALIGN=MIDDLE  ALT=\"+%28cot%28A%29+%2B+1%29%2F+%28cot%28A%29+-+1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find cot(15°): If we realize that 15 = 45-30 then we can use the formula above by setting A = 30:<BR>\n" );
document.write( "cot(15°) = cot(45°-30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28cot%2830%29+%2B+1%29%2F+%28cot%2830%29+-+1%29+ ALIGN=MIDDLE  ALT=\"+%28cot%2830%29+%2B+1%29%2F+%28cot%2830%29+-+1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%2830%29+=+cos%2830%29%2Fsin%2830%29+=+%28sqrt%283%29%2F2%29%2F%281%2F2%29+=+sqrt%283%29 ALIGN=MIDDLE  ALT=\"cot%2830%29+=+cos%2830%29%2Fsin%2830%29+=+%28sqrt%283%29%2F2%29%2F%281%2F2%29+=+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting this we get:<BR>\n" );
document.write( "cot(15°) = cot(45°-30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sqrt%283%29+%2B+1%29%2F+%28sqrt%283%29+-+1%29+ ALIGN=MIDDLE  ALT=\"+%28sqrt%283%29+%2B+1%29%2F+%28sqrt%283%29+-+1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we rationalize the denominator. To do this we will use the conjugate of the denominator which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29+%2B+1 ALIGN=MIDDLE  ALT=\"sqrt%283%29+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "cot(15°) = cot(45°-30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%283%29+%2B+1%29%2F%28sqrt%283%29+%2B+1%29%29%28sqrt%283%29+%2B+1%29%2F+%28sqrt%283%29+-+1%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying this out (using either FOIL or the patterns for (a+b)(a+b) and (a+b)(a-b)) we get:<BR>\n" );
document.write( "cot(15°) = cot(45°-30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28sqrt%283%29%29%5E2+%2B+2sqrt%283%29+%2B+1%5E2%29%2F%28%28sqrt%283%29%29%5E2+-+1%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "cot(15°) = cot(45°-30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283+%2B+2sqrt%283%29+%2B+1%29%2F%283+-+1%29 ALIGN=MIDDLE  ALT=\"%283+%2B+2sqrt%283%29+%2B+1%29%2F%283+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "See how the conjugate made the denominator rational?<BR>\n" );
document.write( "cot(15°) = cot(45°-30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284+%2B+2sqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"%284+%2B+2sqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Reduce the fraction by factoring out 2 and canceling:<BR>\n" );
document.write( "cot(15°) = cot(45°-30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%282+%2B+sqrt%283%29%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"%282%282+%2B+sqrt%283%29%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "cot(15°) = cot(45°-30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2B+sqrt%283%29 ALIGN=MIDDLE  ALT=\"2+%2B+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );