In the course of doing scratchwork to answer this question, I had occasion to write the trigonometric identity $$ \arctan x- \arctan(1-x) = \arctan\left( \frac{1-2x}{x^2-x-1} \right). $$ Now notice that this is $$ \arctan\left( -\frac{d}{dx} \log(x^2-x-1) \right) $$ and $x^2-x-1$ is the monic polynomial of lowest degree with rational coefficients that has the golden ratio $(1+\sqrt{5})/2$ as a root, and that number is $\lim\limits_{n\to\infty} F_{n+1}/F_n$, where $F_n$ is the $n$th Fibonacci number.
Googling indicates that someone has found connections between arctangents and Fibonacci numbers: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibpi.html#section2
Is the fact I state above "known"? What related facts are "known"? (Scare quotes because obviously it is known to those who read this question.)