document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=885010>Question 885010</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if tan theta+cot theta=2,find the value of tan7theta+cot7theta.(Sir please explain how to solve problems which have number between sin,cos,tan and theta like tan7theta) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #534953 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5345)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=534953\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE ANSWER:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%287theta%29%2Bcot%287theta%29=-2 ALIGN=MIDDLE  ALT=\"tan%287theta%29%2Bcot%287theta%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "REQUESTED EXPLANATION (not a direct way to solve this problem):<BR>\n" );
document.write( "When I have to deal with trigonometric functions of sums or multiples of angles,<BR>\n" );
document.write( "I look for a table of trigonometric identities.<BR>\n" );
document.write( "Trigonometric identities are relations (formulas) like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282A%29=2sin%28A%29%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"sin%282A%29=2sin%28A%29%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are many more of those.<BR>\n" );
document.write( "(I do not have those formulas memorized, and<BR>\n" );
document.write( "although I can rediscover trig identity formulas,<BR>\n" );
document.write( "it is much faster to look them up).<BR>\n" );
document.write( "Your teacher may believe otherwise, but as professor Peter Alfeld says in his U. of Utah website,<BR>\n" );
document.write( "\"Understanding mathematics requires no memorization whatsoever\".<BR>\n" );
document.write( "(I never met him or stepped into U. of Utah,<BR>\n" );
document.write( "but I found his website when looking for someone who agreed with my philosophy.<BR>\n" );
document.write( "I do not like to memorize what I can rediscover/re-deduce/calculate by myself.<BR>\n" );
document.write( "What I cannot calculate, like phone numbers,<BR>\n" );
document.write( "if possible, I write down and/or look up.<BR>\n" );
document.write( "I prefer to remember facts and formulas without trying,<BR>\n" );
document.write( "just letting the stuff get ingrained in my memory from repeated use).<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE SOLUTION:<BR>\n" );
document.write( "You are unlikely to find trig identities dealing with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7theta%29%7D%7D+.%0D%0AThe+key+to+solving+the+problem+is+to+realize+that+your+%7B%7B%7Btheta ALIGN=MIDDLE  ALT=\"7theta%29%7D%7D+.%0D%0AThe+key+to+solving+the+problem+is+to+realize+that+your+%7B%7B%7Btheta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a very special angle, that should be very familiar to you.<BR>\n" );
document.write( "You are given <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29%2Bcot%28theta%29=2 ALIGN=MIDDLE  ALT=\"tan%28theta%29%2Bcot%28theta%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=sin%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"tan%28theta%29=sin%28theta%29%2Fcos%28theta%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%28theta%29=cos%28theta%29%2Fsin%28theta%29 ALIGN=MIDDLE  ALT=\"cot%28theta%29=cos%28theta%29%2Fsin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%28theta%29=1%2Ftan%28theta%29 ALIGN=MIDDLE  ALT=\"cot%28theta%29=1%2Ftan%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Those formulas appear in listings of trig identities,<BR>\n" );
document.write( "but are used so often that I do not need to look them up.<BR>\n" );
document.write( " <BR>\n" );
document.write( "You can use<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29%2Bcot%28theta%29=tan%28theta%29%2B1%2Ftan%28theta%29=2 ALIGN=MIDDLE  ALT=\"tan%28theta%29%2Bcot%28theta%29=tan%28theta%29%2B1%2Ftan%28theta%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=1 ALIGN=MIDDLE  ALT=\"tan%28theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is easy enough, because with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=tan%28theta%29 ALIGN=MIDDLE  ALT=\"x=tan%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > the equation turns into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Fx=2 ALIGN=MIDDLE  ALT=\"x%2B1%2Fx=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B1%29%2Fx=2 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B1%29%2Fx=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B1=2x ALIGN=MIDDLE  ALT=\"x%5E2%2B1=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2B1=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=0 ALIGN=MIDDLE  ALT=\"x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "From there you realize that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F4+%2B-+k%2Api ALIGN=MIDDLE  ALT=\"theta=pi%2F4+%2B-+k%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=45%5Eo+%2B-+k%2A180%5Eo ALIGN=MIDDLE  ALT=\"theta=45%5Eo+%2B-+k%2A180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you like degrees).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You can also work with the sine and cosine trigonometric identity formulas to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2theta ALIGN=MIDDLE  ALT=\"2theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29%2Bcot%28theta%29=sin%28theta%29%2Fcos%28theta%29%2Bcos%28theta%29%2Fsin%28theta%29=%28%28sin%28theta%29%29%5E2%2B%28cos%28theta%29%29%5E2%29%2F%28sin%28theta%29%2Acos%28theta%29%29=1%2F%28sin%28theta%29%2Acos%28theta%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=1%2F%28sin%28theta%29%2Acos%28theta%29%29 ALIGN=MIDDLE  ALT=\"2=1%2F%28sin%28theta%29%2Acos%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28theta%29%2Acos%28theta%29=1 ALIGN=MIDDLE  ALT=\"2sin%28theta%29%2Acos%28theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now, we use the trig identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282A%29=2sin%28A%29%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"sin%282A%29=2sin%28A%29%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28theta%29%2Acos%28theta%29=1 ALIGN=MIDDLE  ALT=\"2sin%28theta%29%2Acos%28theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29=1 ALIGN=MIDDLE  ALT=\"sin%282theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The angle between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%220%22 ALIGN=MIDDLE  ALT=\"%220%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > whose sine is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo ALIGN=MIDDLE  ALT=\"90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you like degrees better than radians),<BR>\n" );
document.write( "but all angles coterminal with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > have a sine of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2theta=pi%2F2+%2B-+2k%2Api ALIGN=MIDDLE  ALT=\"2theta=pi%2F2+%2B-+2k%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F4+%2B-+k%2Api ALIGN=MIDDLE  ALT=\"theta=pi%2F4+%2B-+k%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=45%5Eo+%2B-+k%2A180%5Eo ALIGN=MIDDLE  ALT=\"theta=45%5Eo+%2B-+k%2A180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you like degrees).<BR>\n" );
document.write( " <BR>\n" );
document.write( "Any way you get there,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F4+%2B-+k%2Api ALIGN=MIDDLE  ALT=\"theta=pi%2F4+%2B-+k%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=45%5Eo+%2B-+k%2A180%5Eo ALIGN=MIDDLE  ALT=\"theta=45%5Eo+%2B-+k%2A180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you like degrees).<BR>\n" );
document.write( "Since tangent and cotangent have a period of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "tangent and cotangent have the same value for all those angles.<BR>\n" );
document.write( "They also have the same value for all angles represented by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7theta=7pi%2F4+%2B-+7k%2Api ALIGN=MIDDLE  ALT=\"7theta=7pi%2F4+%2B-+7k%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so we can work with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=7pi%2F4-2pi=-pi%2F4 ALIGN=MIDDLE  ALT=\"theta=7pi%2F4-2pi=-pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-45%5Eo ALIGN=MIDDLE  ALT=\"-45%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you like).<BR>\n" );
document.write( " <BR>\n" );
document.write( "You may know that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F4%29=sqrt%282%29%2F2=cos%28pi%2F4%29 ALIGN=MIDDLE  ALT=\"sin%28pi%2F4%29=sqrt%282%29%2F2=cos%28pi%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28pi%2F4%29=1 ALIGN=MIDDLE  ALT=\"tan%28pi%2F4%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28-A%29=-tan%28A%29%7D%7D+and+%7B%7B%7Bcot%28-A%29=-cot%28A%29 ALIGN=MIDDLE  ALT=\"tan%28-A%29=-tan%28A%29%7D%7D+and+%7B%7B%7Bcot%28-A%29=-cot%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "You can use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28-pi%2F4%29=-tan%28pi%2F4%29=-1 ALIGN=MIDDLE  ALT=\"tan%28-pi%2F4%29=-tan%28pi%2F4%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%28-pi%2F4%29=-cot%28pi%2F4%29=-1 ALIGN=MIDDLE  ALT=\"cot%28-pi%2F4%29=-cot%28pi%2F4%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to calculate that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%287theta%29%2Bcot%287theta%29=-1%2B%28-1%29=-2 ALIGN=MIDDLE  ALT=\"tan%287theta%29%2Bcot%287theta%29=-1%2B%28-1%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "You can also use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28-A%29=-tan%28A%29%7D%7D+and+%7B%7B%7Bcot%28-A%29=-cot%28A%29 ALIGN=MIDDLE  ALT=\"tan%28-A%29=-tan%28A%29%7D%7D+and+%7B%7B%7Bcot%28-A%29=-cot%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to justify that,<BR>\n" );
document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7theta=7pi%2F4+%2B-+7k%2Api ALIGN=MIDDLE  ALT=\"7theta=7pi%2F4+%2B-+7k%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > is coterminal with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-theta ALIGN=MIDDLE  ALT=\"-theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29%2Bcot%28theta%29=2 ALIGN=MIDDLE  ALT=\"tan%28theta%29%2Bcot%28theta%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > implies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%287theta%29%2Bcot%287theta%29=-2 ALIGN=MIDDLE  ALT=\"tan%287theta%29%2Bcot%287theta%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );