$\arctan(1/2)$ seems to be some strange, irrational angle, and the same goes for $\arctan(1/3)$, but those two angles seem to sum up to $45$ degrees. This seems like a mystery to me even though I can derive the result algebraically as follows, by using the summation formula for the tangent function.
$\tan\big(\arctan(1/2)+\arctan (1/3)\big)=\dfrac{5/6}{1 – 1/6}=1\,.$
Can someone come up with a geometric explanation?
A remark: I started thinking about the described arctan puzzle while I was trying to solve a problem in complex analysis, namely this one:
Best Answer
There are many good geometric renderings; my favorite invokes the familiar Red Cross symbol. This symbol consists of a central square block sharing each of its edges with an additional congruent block. We superpose right triangle $ABC$ and label some additional vertices $D,E,F$ as shown below.
$\triangle \space ACD$ and $CBE$ are right triangles with congruent legs, so congruent triangles by SAS; thus their hypotenuses which are also legs of right $\triangle ABC$ are congruent. So the acute angle $BAC$ measures $45°$. But the component of that angle within right $\triangle ACD$ measures $\arctan(1/2)$ and the remaining component within right $\triangle ABF$ measures $\arctan(1/3)$. Thereby $\arctan(1/2)+\arctan(1/3)=45°$.