document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=629265>Question 629265</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let tan(alpha) , tan(beta) be the roots of equation x^2-px+q=0, then find cos2(alpha + beta).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #396454 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=396454\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There may be be a much easier way to do this. But since you haven't gotten a response in over a day, I'd thought I'd give it a try.<br><BR>\n" );
document.write( "To save on typing I'm going to use A and B instead of alpha and beta.<br><BR>\n" );
document.write( "Using the Quadratic Formula on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-px%2Bq=0 ALIGN=MIDDLE  ALT=\"x%5E2-px%2Bq=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-p%29+%2B-+sqrt%28%28-p%29%5E2-4%281%29%28q%29%29%29%2F2%281%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-p%29+%2B-+sqrt%28%28-p%29%5E2-4%281%29%28q%29%29%29%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-p%29+%2B-+sqrt%28p%5E2-4%281%29%28q%29%29%29%2F2%281%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-p%29+%2B-+sqrt%28p%5E2-4%281%29%28q%29%29%29%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-p%29+%2B-+sqrt%28p%5E2-4q%29%29%2F2%281%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-p%29+%2B-+sqrt%28p%5E2-4q%29%29%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28p+%2B-+sqrt%28p%5E2-4q%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%28p+%2B-+sqrt%28p%5E2-4q%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is short for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28p+%2B+sqrt%28p%5E2-4q%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%28p+%2B+sqrt%28p%5E2-4q%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28p+-+sqrt%28p%5E2-4q%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%28p+-+sqrt%28p%5E2-4q%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "It doesn't matter which one we say is tan(A) and which one is tan(B). So I'll make the first root tan(A) and the second one tan(B).<br><BR>\n" );
document.write( "Next we use the formula for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%2BB%29+=+%28tan%28A%29+%2B+tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29 ALIGN=MIDDLE  ALT=\"tan%28A%2BB%29+=+%28tan%28A%29+%2B+tan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Inserting our roots into this formula we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%2BB%29+=+%28%28%28p+%2B+sqrt%28p%5E2-4q%29%29%2F2%29+%2B+%28%28p+-+sqrt%28p%5E2-4q%29%29%2F2%29%29%2F%281-%28%28p+%2B+sqrt%28p%5E2-4q%29%29%2F2%29%2A%28%28p+-+sqrt%28p%5E2-4q%29%29%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying... (Note: to multiply the two numerators in the denominator, I am going to use the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > pattern.)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%2BB%29+=+%28%282p%29%2F2%29%2F%281-%28%28p%29%5E2+-+%28sqrt%28p%5E2-4q%29%29%5E2%29%2F4%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=tan%28A%2BB%29+=+p%2F%281-%28p%5E2+-+%28p%5E2-4q%29%29%2F4%29 ALIGN=MIDDLE  ALT=\"tan%28A%2BB%29+=+p%2F%281-%28p%5E2+-+%28p%5E2-4q%29%29%2F4%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%28A%2BB%29+=+p%2F%281-%284q%2F4%29%29 ALIGN=MIDDLE  ALT=\"tan%28A%2BB%29+=+p%2F%281-%284q%2F4%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%28A%2BB%29+=+p%2F%281-q%29 ALIGN=MIDDLE  ALT=\"tan%28A%2BB%29+=+p%2F%281-q%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "From this we are going to find an expression for sin(A+B). tan is opposite/adjacent. So draw a right triangle, pick one of the acute angles and make the opposite side \"p\" and the adjacent side \"1+q\". Then use the Pythagorean Theorem to find the hypotenuse. You should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28p%5E2+%2B+%281%2Bq%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28p%5E2+%2B+%281%2Bq%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since sin is opposite/hypotenuse:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%2BB%29+=+p%2Fsqrt%28p%5E2+%2B+%281%2Bq%29%5E2%29 ALIGN=MIDDLE  ALT=\"sin%28A%2BB%29+=+p%2Fsqrt%28p%5E2+%2B+%281%2Bq%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Next we will use the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282x%29+=+1+-+2sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"cos%282x%29+=+1+-+2sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > variation of the cos(2x) formulas to find our answer. Replacing the x's with (A+B)'s:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282%28a%2BB%29%29+=+1+-+2sin%5E2%28A%2BB%29 ALIGN=MIDDLE  ALT=\"cos%282%28a%2BB%29%29+=+1+-+2sin%5E2%28A%2BB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Replacing the sin(A+B) with the expression we found earlier:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282%28a%2BB%29%29+=+1+-+2%28p%2Fsqrt%28p%5E2+%2B+%281%2Bq%29%5E2%29%29%5E2 ALIGN=MIDDLE  ALT=\"cos%282%28a%2BB%29%29+=+1+-+2%28p%2Fsqrt%28p%5E2+%2B+%281%2Bq%29%5E2%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282%28a%2BB%29%29+=+1+-+2%28p%5E2%2F%28p%5E2+%2B+%281%2Bq%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"cos%282%28a%2BB%29%29+=+1+-+2%28p%5E2%2F%28p%5E2+%2B+%281%2Bq%29%5E2%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=cos%282%28a%2BB%29%29+=+1+-+%282p%5E2%29%2F%28p%5E2+%2B+%281%2Bq%29%5E2%29 ALIGN=MIDDLE  ALT=\"cos%282%28a%2BB%29%29+=+1+-+%282p%5E2%29%2F%28p%5E2+%2B+%281%2Bq%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282%28a%2BB%29%29+=+%28p%5E2+%2B+%281%2Bq%29%5E2%29%2F%28p%5E2+%2B+%281%2Bq%29%5E2%29+-+%282p%5E2%29%2F%28p%5E2+%2B+%281%2Bq%29%5E2%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=cos%282%28a%2BB%29%29+=+%28-p%5E2+%2B+%281%2Bq%29%5E2%29%2F%28p%5E2+%2B+%281%2Bq%29%5E2%29 ALIGN=MIDDLE  ALT=\"cos%282%28a%2BB%29%29+=+%28-p%5E2+%2B+%281%2Bq%29%5E2%29%2F%28p%5E2+%2B+%281%2Bq%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );