document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=942249>Question 942249</A>: <I><SPAN STYLE=\"word-break: break-all;\">  tan(&#952;)=cot(4&#952;+20)   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854627 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53937)</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=854627\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "tan(x)=cot(4x+20°).  Find x.<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution in the post by @lwsshar3 is a part of the truth, but not the whole truth.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution by @lwsshar3 is correct if and only if both angles, x and 4x+20° are in QI.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;But if we look for all possible angles x between 0° and 180°  (180° is the period for both functions\r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in the left and the right side of the equation, so it is NATURAL to investigate this interval), \r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;then his solution is INCOMPLETE, because there are 5 (five) different angles 'x' satisfying the given equation \r<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in interval [0°,180°).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;And this is the whole truth.  See my solution below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Equation\r\n" );
document.write( "\r\n" );
document.write( "    tan(x) = cot(4x+20°)    (1)\r\n" );
document.write( "\r\n" );
document.write( "implies\r\n" );
document.write( "\r\n" );
document.write( "    x + (4x+20°) = 90° + k*180°, k = 0, +/-1, +/-2, . . . (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "because both functions tan and cot are periodical with period <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we can rewrite equation (2) in the form\r\n" );
document.write( "\r\n" );
document.write( "    5x = 70° + k*180°, k = 0, 1, 2, 3, 4.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We can limit ourselves with 5 values of k, because for other values of 'k', the angles 'x' will be geometrically the same,\r\n" );
document.write( "but formally will be outside of the interval [0°,180°).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For these 5 values of 'k', we will have 5 different values of x in the interval [0°,180°)\r\n" );
document.write( "\r\n" );
document.write( "    x = 14°,  14°+36° = 50°,  14°+72° = 86°,  14°+108° = 122°,  14°+144° = 158°.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "<U>ANSWER</U>. Taking into account the periodicity of the left side and the right side of the given equation, \r\n" );
document.write( "\r\n" );
document.write( "it is natural to look for solutions 'x' in the interval [0°,180°).\r\n" );
document.write( "\r\n" );
document.write( "In this interval, the given equation has 5 (five) different solutions 14°, 50°, 86°, 122° and 158°.\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "Solved completely.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "//////////// I N T E R E S T I N G ////////////\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Driven by my curiosity, I posted today (May 21, 2026, about 2:00 PM) this problem to Google Overview Artificial Intelligence<BR>\n" );
document.write( "to look how it treats the problem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It treats it precisely as in the post by @lwsshar3 - so, its analysis is incomplete.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Does it embarrass me? - - - Both \"Yes\" and \"No\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It simply demonstrates the level and the style of work of contemporary AI.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It can not think as a human can (because it is NOT ABLE to think as a human can), <BR>\n" );
document.write( "but simply re-writes from the storage of existing solutions, with no thinking.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );