FT8: Recurrence Relations for Ftan and Fcot

This is the eighth in a series of posts on Fibonacci trigonometry:

Part 1: Basic Definitions

Part 2: The Canonical Basis and Closed-Form Expressions

Part 3: The Basic Analogies

Part 4: Fibonacci Formulas Using sinh, etc. — Real Number Version

Part 5: Fibonacci Formulas Using sinh, etc. — Complex Number Version

Part 6: Trigonometric-Style Identities

Part 7: The Fibonacci Tangent and Cotangent Functions

Part 8: Recurrence Relations for Ftan and Fcot

 

The addition formulas for Ftan and Fcot give us the following recurrence relations:

• 5 Ftan(n + 1) = 1 + 4 / (1 + 5 Ftan(n))

• Fcot(n + 1) = 1 + 4 / (1 + Fcot(n))

These recurrence relations yield continued-fraction-style expansions for Ftan(n) and Fcot(n).

Now, as n approaches infinity, Ftan(n) approaches 1/√5, and Fcot(n) approaches √5. These facts can be seen either from the continued-fraction expansions or from the earlier formulas involving φ.