A Trigonometric Identity for Orbital Eccentricity: cos(2γ) = −e Derived from a Right Triangle Inscribed in a Semicircle

We present an exact trigonometric identity for Keplerian elliptical orbits: define the angle γ by tan(γ) = √((1 + e)/(1 − e)); then cos(2γ) = −e. The relation emerges naturally from Euclidean geometry. Inscribe a right triangle in a semicircle with the diameter as hypotenuse (Thales’ theorem ensures a right angle at the arc vertex). The tangent of the base angle is the ratio of opposite to adjacent legs. By the geometric mean theorem (altitude to the hypotenuse), this ratio takes the square-root form tan(γ) = √(ra / rp). Identifying ra and rp with apoastron and periastron distances yields the orbital expression. The double-angle formula then produces cos(2γ) = −e algebraically. The identity holds for all 0 ≤ e < 1 and is verified numerically on Solar System bodies and exoplanets (NASA Exoplanet Archive). It offers a geometrically grounded angular parametrization of eccentricity useful for orbital analysis and population studies.