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Trigonometry without a word

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Prove the identities:

\[\begin{gathered} \arctan\frac12+\arctan\frac13=\frac\pi4,\\ \arctan1+\arctan2+\arctan3=\pi \end{gathered}\]
Solution
Two geometric proofs on a unit grid. The small right triangle has equal legs, and its 45-degree angle splits into angles with tangents 1/2 and 1/3. In the larger triangle, a perpendicular segment is one third of the upper side. The three angles beside the straight diagonal have tangents 1, 2, and 3. Matching ticks mark equal lengths.
A proof without words, on a unit grid.
Another solution

Let \(\alpha=\arctan(1/2)\) and \(\beta=\arctan(1/3)\). Both lie in \((0,\pi/4)\), and

\[\tan(\alpha+\beta) =\frac{\frac12+\frac13}{1-\frac16}=1.\]

Since \(0<\alpha+\beta<\pi/2\), this gives \(\alpha+\beta=\pi/4\).

For \(t>0\), \(\arctan t+\arctan(1/t)=\pi/2\). Consequently, the second sum equals

\[\frac\pi4+\pi-(\alpha+\beta)=\pi.\]

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