document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=624980>Question 624980</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question 624957: Verify the trig identity: sin^4(x)+cos^4 (x)=3/4 + 1/4 cos(4x). \r<BR>\n" );
document.write( "\n" );
document.write( "How about this solution (different and faster than from jsmallt9):<BR>\n" );
document.write( "sin^4(x)+cos^4(x) = ((sin^2(x) + cos^2(x)))^2 - 2sin^2(x)cos^2(x) = <BR>\n" );
document.write( "1 - 2sin^2(x)cos^2(x) = 1 - (1/2)sin^2(2x) = 1 - (1 - cos(4x))/4 = <BR>\n" );
document.write( "1 - 1/4 + cos(4x)/4 = 3/4 + cos(4x)/4  Done </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #393252 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=393252\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This solution works, too.<br><BR>\n" );
document.write( "The main reason it appears faster than my solutions is that many intermediate steps are being omitted. If I had done this my first solution would look like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28%281-cos%282x%29%29%2F2%29%29%5E4%2B%28sqrt%28%281%2Bcos%282x%29%29%2F2%29%29%5E4=3%2F4+%2B+%281%2F4%29cos%284x%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=2%2F4%2B%282cos%5E2%282x%29%29%2F4+=+3%2F4+%2B+%281%2F4%29cos%284x%29 ALIGN=MIDDLE  ALT=\"2%2F4%2B%282cos%5E2%282x%29%29%2F4+=+3%2F4+%2B+%281%2F4%29cos%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4%2B%281%2F4%29cos%284x%29+=+3%2F4+%2B+%281%2F4%29cos%284x%29 ALIGN=MIDDLE  ALT=\"3%2F4%2B%281%2F4%29cos%284x%29+=+3%2F4+%2B+%281%2F4%29cos%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And we're finished!<br><BR>\n" );
document.write( "And the second solution would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%5E2%28x%29%29%5E2%2B%28cos%5E2%28x%29%29%5E2=3%2F4+%2B+%281%2F4%29cos%284x%29 ALIGN=MIDDLE  ALT=\"%28sin%5E2%28x%29%29%5E2%2B%28cos%5E2%28x%29%29%5E2=3%2F4+%2B+%281%2F4%29cos%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-1%2F2%29cos%282x%29+%2B+1%2F2%29%5E2%2B%28%281%2F2%29cos%282x%29+%2B+1%2F2%29%5E2=3%2F4+%2B+%281%2F4%29cos%284x%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=%281%2F2%29cos%5E2%282x%29+%2B+1%2F2+=+3%2F4+%2B+%281%2F4%29cos%284x%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29cos%5E2%282x%29+%2B+1%2F2+=+3%2F4+%2B+%281%2F4%29cos%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28%281%2F2%29cos%284x%29+%2B+1%2F2%29++%2B+1%2F2+=+3%2F4+%2B+%281%2F4%29cos%284x%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=%281%2F4%29cos%284x%29+%2B+3%2F4+=+3%2F4+%2B+%281%2F4%29cos%284x%29 ALIGN=MIDDLE  ALT=\"%281%2F4%29cos%284x%29+%2B+3%2F4+=+3%2F4+%2B+%281%2F4%29cos%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );