CCFU Proof 7 — Arctangent Identity: arctan(1/φ) + arctan(1/φ³) = π/4

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CCFU Proof 7 — Arctangent Identity: arctan(1/φ) + arctan(1/φ³) = π/4

05/15/2026

Given.
Let φ = (1+√5)/2. Then φ² = φ + 1 and φ > 1.

Claim.

    arctan(1/φ) + arctan(1/φ³) = π/4

Proof.
Using the addition formula:

    arctan(a) + arctan(b) = arctan((a+b)/(1−ab))  when ab < 1

Let a = 1/φ, b = 1/φ³.

Branch safety.
ab = 1/φ⁴ < 1. Since a > 0, b > 0, and ab < 1,
the sum lies in (0, π/2), so no branch correction is needed.

Numerator:

    a + b = 1/φ + 1/φ³ = (φ² + 1)/φ³ = (φ + 2)/φ³

Denominator:

    1 − ab = 1 − 1/φ⁴ = (φ⁴ − 1)/φ⁴

Powers of φ.  Using φ² = φ + 1:

    φ³ = 2φ + 1
    φ⁴ = 3φ + 2
    φ⁴ − 1 = 3φ + 1

Ratio:

    (a+b)/(1−ab) = [(φ+2)/φ³] / [(3φ+1)/φ⁴]
                 = φ(φ+2)/(3φ+1)
                 = (φ²+2φ)/(3φ+1)
                 = (φ+1+2φ)/(3φ+1)
                 = (3φ+1)/(3φ+1)
                 = 1

Therefore arctan(1/φ) + arctan(1/φ³) = arctan(1) = π/4.  ∎