document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=188287>Question 188287</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let f(x) = cos x.  Prove that...<BR>\n" );
document.write( "[f(x+h) - f(x)]/h = cosx[(cosh-1)/h]- sinx[sinh/h]<BR>\n" );
document.write( "I have no clue how to do this one. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #141137 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=141137\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%2Bh%29=cos%28x%29cos%28h%29-sin%28x%29sin%28h%29 ALIGN=MIDDLE  ALT=\"cos%28x%2Bh%29=cos%28x%29cos%28h%29-sin%28x%29sin%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > this is one of the many trig identities\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+cos%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29=cos%28x%2Bh%29=cos%28x%29cos%28h%29-sin%28x%29sin%28h%29 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29=cos%28x%2Bh%29=cos%28x%29cos%28h%29-sin%28x%29sin%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or simply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29=cos%28x%29cos%28h%29-sin%28x%29sin%28h%29 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29=cos%28x%29cos%28h%29-sin%28x%29sin%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28+cos%28x%29cos%28h%29-sin%28x%29sin%28h%29+%29-+cos%28x%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28%28+cos%28x%29cos%28h%29-sin%28x%29sin%28h%29+%29-+cos%28x%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+cos%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Bh%29=cos%28x%29cos%28h%29-sin%28x%29sin%28h%29 ALIGN=MIDDLE  ALT=\"f%28x%2Bh%29=cos%28x%29cos%28h%29-sin%28x%29sin%28h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29cos%28h%29+-+cos%28x%29+-sin%28x%29sin%28h%29+%29%2Fh+ ALIGN=MIDDLE  ALT=\"%28cos%28x%29cos%28h%29+-+cos%28x%29+-sin%28x%29sin%28h%29+%29%2Fh+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Remove the parenthesis and rearrange the terms (I just moved the last cosine term right after the first term)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29+%28cos%28h%29+-+1%29+-sin%28x%29sin%28h%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28cos%28x%29+%28cos%28h%29+-+1%29+-sin%28x%29sin%28h%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the first two terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29+%28cos%28h%29+-+1%29%29%2Fh+-%28sin%28x%29sin%28h%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28cos%28x%29+%28cos%28h%29+-+1%29%29%2Fh+-%28sin%28x%29sin%28h%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh=%28cos%28x%29+%28cos%28h%29+-+1%29%29%2Fh+-%28sin%28x%29sin%28h%29%29%2Fh ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=cos%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: if you are in Calculus, then this would show that the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sin%28x%29 ALIGN=MIDDLE  ALT=\"-sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with a few more steps). </SPAN>\n" );
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