Sum of three arctangents

Text

Prove that $\arctan 1+\arctan 2+\arctan 3=\pi$.

Solution

We have $\arctan 1=\frac{\pi}{4}$. For the other two, by the addition formula: $\tan(\arctan 2+\arctan 3)=\frac{2+3}{1-2\cdot 3}=\frac{5}{-5}=-1.$ Since $\arctan 2,\arctan 3\in\left(\frac{\pi}{4},\frac{\pi}{2}\right)$, their sum lies in $\left(\frac{\pi}{2},\pi\right)$; the only angle in that interval with tangent $-1$ is $\frac{3\pi}{4}$. Therefore $\arctan 1+\arctan 2+\arctan 3=\frac{\pi}{4}+\frac{3\pi}{4}=\boxed{\pi}\approx 3.1416\ \text{rad}.$