The purpose of this paper is both to observe, understand and appreciate the link between the Fibonacci sequence and the ubiquitous mathematical constant, $\pi$. It proves the following series for $\pi$, making use of the Fibonacci numbers.
where $F_n$ is the nth Fibonacci number.
The derivation of this identity employs several different mathematical concepts.
Angle Sum Formulae
We begin with the well-known trigonometric sum formulae:
By dividing these expressions, we obtain a corresponding formula for the tangent: