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ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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Morrie's law: cos⁡(π/9)cos⁡(2π/9)cos⁡(4π/9)=1/8\cos(\pi/9)\cos(2\pi/9)\cos(4\pi/9)=1/8

Example

cos⁡(π/9)cos⁡(2π/9)cos⁡(4π/9)=1/8.\cos(\pi/9)\cos(2\pi/9)\cos(4\pi/9)=1/8. The conventions and prerequisite facts used below are recorded in Double-angle and quadratic power-reduction identities, Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions, The zero sets of sine and cosine and the least positive common period 2 pi.

Facts & Assumptions

Given: x=π/9x=\pi/9.

Verification

1.1

Apply sin⁡(2t)=2sin⁡tcos⁡t\sin(2t)=2\sin t\cos t successively at t=xt=x, 2x2x, and 4x4x. This gives 8sin⁡xcos⁡xcos⁡2xcos⁡4x=sin⁡8x8\sin x\cos x\cos2x\cos4x=\sin8x.

algebra
2.1

Here 8x=π−x8x=\pi-x, so the supplementary identity gives sin⁡8x=sin⁡x\sin8x=\sin x. Since 0<x<π0<x<\pi, the sine-zero characterization gives sin⁡x≠0\sin x\ne0. Cancel it in step 1.1 to obtain 8cos⁡xcos⁡2xcos⁡4x=18\cos x\cos2x\cos4x=1.

step 1.1given∎

Depends on

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