Here’s an unusual formula for π. Let Fn be the nth Fibonacci number. Then
As mysterious as this equation may seem, it’s not hard to prove. The arctangent identity
shows that the sum telescopes, leaving only the first term, arctan(1) = π/4. To prove the arctangent identity, take the tangent of both sides, use the addition law for tangents, and use the Fibonacci identity
See this post for an even more remarkable formula relating Fibonacci numbers and π.