Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?
Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?
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0http://www.math.hmc.edu/funfacts/ffiles/20005.2.shtml – 2012-09-16
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0It is true in general that $\tan^{-1}(a) + \tan^{-1}(b) + \tan^{-1}(c) = \pi$ when $a+b+c=abc$ (and $a,b,c$ positive). This is the converse of what is proved [here](http://math.stackexchange.com/questions/477364/prove-that-tan-a-tan-b-tan-c-tan-a-tan-b-tan-c-abc-180-circ). – 2015-10-19
5 Answers
Note that $$ \tan \left(\arctan(1+z) + \arctan\left(2 + z + z^2 \right) + \arctan \left( 3+3\,z+4\,{z}^{2}+2\,{z}^{3}+{z}^{4} \right) \right)=z $$ so that $$\arctan(1+z) + \arctan\left(2 + z + z^2 \right) + \arctan \left( 3+3\,z+4\,{z}^{2}+2\,{z}^{3}+{z}^{4} \right) = \arctan(z) + n \pi $$ for the appropriate integer $n$. For integers $z$ we get interesting arctan identities from this.
$$\eqalign{ \arctan(1) + \arctan\left(2\right)+ \arctan\left(3\right) &= \pi \cr \arctan(2) + \arctan(4) + \arctan(13) &= \arctan(1) + \pi \cr \arctan(3) + \arctan(8) + \arctan(57) &= \arctan(2) + \pi \cr \arctan(4) + \arctan(14) + \arctan(183) &= \arctan(3) + \pi \cr}$$ etc.
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2How did you manage the first equation? – 2012-09-16
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3I asked Maple for integer solutions of $$\frac{a+b+c-abc}{1-ab-bc-ca} = d$$ – 2012-09-16
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0I know that this an old answer, but would you please explain how the integer solutions of $$\frac{a+b+c-abc}{1-ab-bc-ca} = d$$ relate to your first identity? I can't see it on my own :( – 2014-07-23
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0Maple gave me a parametric solution which simplifies to $$\left\{ a={{\it \_Z1}}^{4}+2\,{{\it \_Z1}}^{3}+4\,{{\it \_Z1}}^{2}+3 \,{\it \_Z1}+3,b={{\it \_Z1}}^{2}+{\it \_Z1}+2,c=1+{\it \_Z1},d={\it \_Z1} \right\} $$ – 2014-07-23
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0@math.n00b $\tan(\arctan(a)+\arctan(b)+\arctan(c)) = ?$ – 2016-11-05
Consider $O=(0,0)$, $A=(1,1)$, $B=(-1,3)$, $D=(1,-3)$, $E=(1,0)$.
\begin{align} 2 &= \frac{AB}{AO} = \tan \angle AOB \\ 1 &= \frac{AE}{EO} = \tan \angle AOE \\ 3 &= \frac{DE}{DO} = \tan \angle DOE \end{align}
The points B, O and D are collinear, i.e. $\angle BOD = \tan^{-1}2+\tan^{-1}1+\tan^{-1}3 = \pi$.
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2+1, I like showing facts in drawings - Somehow it makes facts easier to grasp. It may be nicer to explain the part: "The points B, O and E are collinear,..." a bit. Thanks. – 2012-09-16
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1@EmmadKareem: Sorry, it should be B, O, **D** are collinear, and it is pretty obvious that they all fall on the same line $y = -3x$. – 2012-09-16
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2Beautiful. Well done. – 2012-09-16
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2Can I ask how you made that picture? It looks really intuitive. :) – 2012-09-16
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1@David: [GeoGebra](http://www.geogebra.org/cms/). – 2012-09-16
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0You can also rotate around the OED triangle so that it's O(-1,0)B, if you want – 2015-04-13
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0@columbus8myhw that would be [JASON's answer](http://math.stackexchange.com/a/254427/171). – 2015-04-13
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0@kennytm Only if you switch the $\tan^{-1}1$ and $\tan^{-1}2$ around. – 2015-04-13
Proof without word
$\tan^{-1} 1+\tan^{-1} 2+\tan^{-1} 3 =\pi$.
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2Oh! I answered a duplicated question and here is the same nice solution (compare [this](http://math.stackexchange.com/questions/272208/proving-arctan1-arctan2-arctan3-pi/272250#272250)). I don't know why this has only 4 upvotes :-| +1! – 2013-01-07
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1Agree with dtldarek, and trying to fix the upvote count. – 2013-01-07
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0How do you do geometric Proofs? – 2018-07-27
The simplest way is by using complex numbers. It is a trivial computation to show that $$(1+i)(1+2i)(1+3i)=-10$$ Now recall the geometric description of complex multiplication (multiply the lengths and add the angles), and take the argument on both sides of this equation. This gives $$\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$$
$$\tan^{-1}(2)+\tan^{-1}(3)=\tan^{-1}{\left(\frac{2+3}{1-2\cdot 3}\right)}=\tan^{-1}(-1)=n\pi-\frac \pi 4,$$ where $n$ is any integer.
Now the principal value of $\tan^{-1}(x)$ lies in $[-\frac \pi 2, \frac \pi 2]$ precisely in $(0, \frac \pi 2)$ if finite $x>0$. So, the principal value of $\tan^{-1}(2)+\tan^{-1}(3)$ will lie in $(0, \pi) $.
So, the principal value of $\tan^{-1}(2)+\tan^{-1}(3)$ will be $\frac {3\pi} 4$.
Interestingly, the principal value of $\tan^{-1}(-1)$ is $-\frac {\pi} 4$.
But the general values of $\tan^{-1}(2)+\tan^{-1}(3)$ and $\tan^{-1}(-1)$ are same.
Alternatively, $$\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\tan^{-1}{\left(\frac{1+2+3-1\cdot 2\cdot 3}{1-1\cdot 2- 2\cdot 3 -3\cdot 1}\right)}=\tan^{-1}(0)=m\pi$$, where $m$ is any integer.
Now the principal value of $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)$ will lie in $(0 ,\frac {3\pi} 2)$ which is $\pi$.
The principal value of $\tan^{-1}(0)$ is $0\neq \pi$.
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0But typing the value in calculator gives π and not 0. – 2018-04-15
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0@suiz, $$\tan^{-1}(2)+\tan^{-1}(3)$$ will be $$\frac {3\pi} 4$$ right? – 2018-04-15
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0Yes. And arctan(1) gives π/4. If we add it with 3π/4, we get π. Am I wrong with this? – 2018-04-15
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0@suiz, You are correct. See also: https://math.stackexchange.com/questions/523625/showing-arctan-frac23-frac12-arctan-frac125/523626#523626 – 2018-04-15
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0Also, xy >0 in this case. Then how could your writing the value 0 be right? Am I missing something out? – 2018-04-15
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0@suiz, I think you have meant $$(0,\pi)$$ right? Then $($ mean open interval – 2018-04-15