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ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-02
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Morrie's law: cos⁡(π/9)cos⁡(2π/9)cos⁡(4π/9)=1/8

Example

cos⁡(π/9)cos⁡(2π/9)cos⁡(4π/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/9.

Verification

1.1

Apply sin⁡(2t)=2sin⁡tcos⁡t successively at t=x, 2x, and 4x. This gives 8sin⁡xcos⁡xcos⁡2xcos⁡4x=sin⁡8x.

algebra
2.1

Here 8x=π−x, so the supplementary identity gives sin⁡8x=sin⁡x. Since 0<x<π, the sine-zero characterization gives sin⁡x≠0. Cancel it in step 1.1 to obtain 8cos⁡xcos⁡2xcos⁡4x=1.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources