FT7: Fibonacci Tangent and Cotangent Functions

This is the seventh 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

 

Recall that, since F behaves analogously to the sine function, and L to the cosine function, we followed the usual development of trigonometry by defining:

• Ftan(n) = F(n) / L(n), the Fibonacci tangent function
• Fcot(n) = L(n) / F(n), the Fibonacci cotangent function (for n ≠ 0)
• Fsec(n) = 1 / L(n), the Fibonacci secant function
• Fcsc(n) = 1 / F(n), the Fibonacci cosecant function (for n ≠ 0)

We'll continue to use the notation F and L, since those symbols are standard, rather than using Fsin and Fcos, respectively.

Just as 1 + tan²x = sec²x and 1 + cot²x = csc²x, we have:

• 1 – 5 Ftan(x)² = 4 (–1)^x Fsec(x)²
• Fcot(x)² – 5 = 4 (–1)^x Fcsc(x)², for x ≠ 0

Ftan and Fcot are odd functions:

• Ftan(–x) = –Ftan(x)
• Fcot(–x) = –Fcot(x), for x ≠ 0

Fsec and Fcsc have the same opposite-sign properties as L and F:

• Fsec(–x) = (–1)^x Fsec(x).
• Fcsc(–x) = (–1)^x+1 Fcsc(x), for x ≠ 0

There are addition, subtraction, and double-argument formulas (the formulas for Fcot only apply where the indicated function values are defined, of course):

• Ftan(x ± y) = (Ftan(x) ± Ftan(y)) / (1 ± 5 Ftan(x) Ftan(y))
• Fcot(x ± y) = (5 ± Fcot(x) Fcot(y)) / (Fcot(x) ± Fcot(y))
• Ftan(2x) = 2 Ftan(x) / (1 + 5 Ftan(x)²)
• Fcot(2x) = (5 + Fcot(x)²) / (2 Fcot(x))