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Re: Simplify[ArcTan[Tan[a] + Tan[b]]]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg120501] Re: Simplify[ArcTan[Tan[a] + Tan[b]]]
  • From: "Oleksandr Rasputinov" <oleksandr_rasputinov at hmamail.com>
  • Date: Tue, 26 Jul 2011 07:07:24 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <j0jk6s$ovj$1@smc.vnet.net>

On Mon, 25 Jul 2011 12:31:40 +0100, rych <rychphd at gmail.com> wrote:

> Can't Mathematica tell anything about ArcTan[Tan[a] + Tan[b]]? Maybe
> there exists no useful identity for that? By useful I mean the
> expression with functions of the "angles" a and b other than tangents.
> Thanks
> Igor

It can certainly tell you something:

In[1] :=
f = Function[{expr},
   100 Count[expr, _Tan, {0, Infinity}] +
   100 Count[expr, _ArcTan, {0, Infinity}] +
   LeafCount[expr]
  ];

In[2] :=
FullSimplify[
   ArcTan[Tan[a] + Tan[b]],
   ComplexityFunction -> f
  ]

Out[2] =
1/2 I (Log[1-I Sec[a] Sec[b] Sin[a+b]]-Log[1+I Sec[a] Sec[b] Sin[a+b]])

However, this result is arguably neither interesting nor useful.

Best,

O. R.




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