document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=980781>Question 980781</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use tan2&#952;= sin2&#952;/cos2&#952; to prove the double angle formula for tangent: tan2&#952;= 2tan&#952;/1-tan^2(theta) can someone please help me with this equation and explain it to me? Thank you.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #601873 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52790)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=601873\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
document.write( "\n" );
document.write( "See the lesson &nbsp;<A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-formulas.lesson>Addition and subtraction formulas</A>&nbsp; in this site,\r<BR>\n" );
document.write( "\n" );
document.write( "section &nbsp;\"Proof of the addition and subtraction formulas for tangents\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Or, &nbsp;below is a direct derivation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We have\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29%2Fcos%282theta%29 ALIGN=MIDDLE  ALT=\"sin%282theta%29%2Fcos%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "according to the definition of tangents as the ratio of sines and cosines.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next, \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29 ALIGN=MIDDLE  ALT=\"sin%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asin%28theta%29%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"2%2Asin%28theta%29%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is the formula for sines of the double argument. &nbsp;Similarly, \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282theta%29 ALIGN=MIDDLE  ALT=\"cos%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28theta%29+-+sin%5E2%28theta%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28theta%29+-+sin%5E2%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is the formula for cosines of the double argument.  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, &nbsp;substitute &nbsp;(2)&nbsp; and &nbsp;(3)&nbsp; into the numerator and the denominator of &nbsp;(1). &nbsp;You will get \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Asin%28theta%29%2Acos%28theta%29%29%2F%28cos%5E2%28theta%29+-+sin%5E2%28theta%29%29 ALIGN=MIDDLE  ALT=\"%282%2Asin%28theta%29%2Acos%28theta%29%29%2F%28cos%5E2%28theta%29+-+sin%5E2%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(4) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, &nbsp;divide the numerator and denominator in the right side of &nbsp;(4)&nbsp; by &nbsp;<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;\" >. &nbsp;You will get \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2A%28sin%28theta%29%2Fcos%28theta%29%29%29%2F%281+-+sin%5E2%28theta%29%2Fcos%5E2%28theta%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(5) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As a last step, &nbsp;replace the ratio &nbsp;<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;\" >&nbsp; by &nbsp;<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;\" >&nbsp; in &nbsp;(5). &nbsp;You will get \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282theta%29 ALIGN=MIDDLE  ALT=\"tan%282theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Atan%28theta%29%2F%281+-+tan%5E2%28theta%29%29 ALIGN=MIDDLE  ALT=\"2%2Atan%28theta%29%2F%281+-+tan%5E2%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(6) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is what has to be proved. &nbsp;You finally got the formula you needed. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );