A Guest Appearance - Futility Closet

The Fibonacci numbers are the ones in this sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Each number is the sum of the two that precede it. But now, interestingly:

$\displaystyle \mathrm{arctan} \left ( \frac{1}{1} \right ) = \mathrm{arctan} \left ( \frac{1}{2} \right ) + \mathrm{arctan} \left ( \frac{1}{3} \right )\\ \mathrm{arctan} \left ( \frac{1}{3} \right ) = \mathrm{arctan} \left ( \frac{1}{5} \right ) + \mathrm{arctan} \left ( \frac{1}{8} \right )\\ \mathrm{arctan} \left ( \frac{1}{8} \right ) = \mathrm{arctan} \left ( \frac{1}{13} \right ) + \mathrm{arctan} \left ( \frac{1}{21} \right )\\ \mathrm{arctan} \left ( \frac{1}{21} \right ) = \mathrm{arctan} \left ( \frac{1}{34} \right ) + \mathrm{arctan} \left ( \frac{1}{55} \right )\\$

“And so on!” writes James Tanton in Mathematics Galore! (2012). “The first relation, for instance, states that a line of slope 1/2 stacked with a line of slope 1/3 gives a line of slope 1. (Can you prove the relations?)”

(Ko Hayashi, “Fibonacci Numbers and the Arctangent Function,” Mathematics Magazine 76:3 [June 2003], 215.)