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

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

Out =
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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