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MathGroup Archive 2006

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simplify a trig expression

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65415] simplify a trig expression
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Fri, 31 Mar 2006 06:09:08 -0500 (EST)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

A direct substitution (with paper and pencil) gives that the integral of 
  Cos[x]/(Sin[x] + 1) is Log[Sin[x] + 1].  This is valid provided Sin[x] 
is not -1.

Mathematica gives:

   Integrate[Cos[x]/(Sin[x] + 1), x]
2 Log[Cos[x/2] + Sin[x/2]]

Is there some simple way to coerce the latter Mathematica-supplied 
result into the paper-and-pencil answer?

The closest I could get is:

   Log[TrigExpand[Expand[(Cos[x/2] + Sin[x/2])^2]]] /.
   {Sin[x/2] -> Sqrt[(1 - Cos[x])/2],
    Cos[x/2] -> Sqrt[(1 + Cos[x])/2]}
Log[1 + Sqrt[1 - Cos[x]]*Sqrt[1 + Cos[x]]]

Am I not seeing some easier TrigExpand or TrigReduce method?

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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