Proof Without Words: A Trigonometric Proof of the Arithmetic Mean–Geometric Mean Inequality Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland, OR Lemma. Given x ∈ (0, π/2), the sum tan x + cot x ≥ 2.

tan x

+ co tx

Proof.

x

x 2 2 tan x

2 cot x

Theorem. Given a, b > 0, their arithmetic mean is greater than their geometric mean. Proof.

a x b

√ √ a b √ + √ ≥ 2, a b

a+b √ ≥ ab. 2

Notes. We have equality in the lemma if and only if x = π/4, hence we have equality in the theorem if and only if a = b. In the first figure, the angle in the middle (blue) triangle at the center of the semicircle has measure 2x, hence tan x + cot x = 2 csc(2x), an identity known as Eisenstein’s duplication formula [1]. The lemma and the theorem are actually equivalent: Setting (a, b) = (tan x, cot x) in the theorem yields the lemma. Summary. We prove wordlessly the arithmetic mean-geometric mean inequality for two positive numbers by an equivalent trigonometric inequality.

Reference 1. L. Tan, Proof without words: Eisenstein’s duplication formula, Math. Mag. 71 (1998) 207, http:// dx.doi.org/10.2307/2691206. http://dx.doi.org/10.4169/college.math.j.46.1.42 MSC: 26D05, 26E60

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