document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=634797>Question 634797</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, this question has completely stumped me.<BR>\n" );
document.write( "Let A,B,C  be the angles of a triangle. Show that tanA+tanB+tanC=tanAtanBtanC\r<BR>\n" );
document.write( "\n" );
document.write( "I've tried using identities to turn it into sin/cos but I got stuck and couldn't go any further.\r<BR>\n" );
document.write( "\n" );
document.write( "I appreciate your help,\r<BR>\n" );
document.write( "\n" );
document.write( "Thankyou </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #399910 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=399910\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The key is the fact that A, B and C are angles of a triangle. So A+B+C = 180.<br><BR>\n" );
document.write( "Solving this equation for C we get:<BR>\n" );
document.write( "C = 180 - (A+B)<BR>\n" );
document.write( "So<BR>\n" );
document.write( "tan(C) = tan(180-(A+B))<BR>\n" );
document.write( "Using the tan(x-y) formula on the right wide we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28C%29+=+%28tan%28180%29-tan%28A%2BB%29%29%2F%281%2Btan%28180%29%2Atan%28A%2BB%29%29 ALIGN=MIDDLE  ALT=\"tan%28C%29+=+%28tan%28180%29-tan%28A%2BB%29%29%2F%281%2Btan%28180%29%2Atan%28A%2BB%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since tan(180) = 0 this simplifies:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28C%29+=+%280-tan%28A%2BB%29%29%2F%281%2B0%2Atan%28A%2BB%29%29 ALIGN=MIDDLE  ALT=\"tan%28C%29+=+%280-tan%28A%2BB%29%29%2F%281%2B0%2Atan%28A%2BB%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28C%29+=+%28-tan%28A%2BB%29%29%2F%281%2B0%29 ALIGN=MIDDLE  ALT=\"tan%28C%29+=+%28-tan%28A%2BB%29%29%2F%281%2B0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28C%29+=+%28-tan%28A%2BB%29%29%2F1 ALIGN=MIDDLE  ALT=\"tan%28C%29+=+%28-tan%28A%2BB%29%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28C%29+=+-tan%28A%2BB%29 ALIGN=MIDDLE  ALT=\"tan%28C%29+=+-tan%28A%2BB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the tan(x+y) formula on the right side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28C%29+=+-%28%28tan%28A%29%2Btan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29 ALIGN=MIDDLE  ALT=\"tan%28C%29+=+-%28%28tan%28A%29%2Btan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28C%29+=+%28-tan%28A%29-tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29 ALIGN=MIDDLE  ALT=\"tan%28C%29+=+%28-tan%28A%29-tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Remember this equation. We will use it now and again later.<br><BR>\n" );
document.write( "We can substitute the above for tan(C) in the left side of the given equation (which we are trying to prove):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%29+%2Btan%28B%29+%2B+%28-tan%28A%29-tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The right side is one term (of three factors). So it will help if we make the left side one term, too. So we'll get common denominators and add:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%29%2A%28%281-tan%28A%29%2Atan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+%2Btan%28B%29%28%281-tan%28A%29%2Atan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+%2B+%28-tan%28A%29-tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28tan%28A%29-tan%5E2%28A%29%2Atan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+%2B%28%28tan%28B%29-tan%28A%29tan%5E2%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+%2B+%28%28-tan%28A%29-tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When we add, some of the terms in the numerators cancel:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28cross%28tan%28A%29%29-tan%5E2%28A%29%2Atan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+%2B%28%28cross%28tan%28B%29%29-tan%28A%29tan%5E2%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+%2B+%28%28cross%28-tan%28A%29%29-cross%28tan%28B%29%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%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=%28-tan%5E2%28A%29%2Atan%28B%29-tan%28A%29tan%5E2%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The right side has factors of tan(A) and tan(B). So does the left side. So lets factor them out:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28tan%28A%29%2Atan%28B%29%2A%28-tan%28A%29-tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which can be rewritten as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%29%2Atan%28B%29%2A%28%28-tan%28A%29-tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And if you look at the third factor and at the equaiton I have asked you to remember, you will see that they are the same. So the third factor is tan(C):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%29%2Atan%28B%29%2Atan%28C%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%29 ALIGN=MIDDLE  ALT=\"tan%28A%29%2Atan%28B%29%2Atan%28C%29+=+tan%28A%29%2Atan%28B%29%2Atan%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );