Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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Triple-angle identities for sine, cosine, and tangent

Statement

For every real xx, sin⁡3x=3sin⁡x−4sin⁡3x,cos⁡3x=4cos⁡3x−3cos⁡x.\sin3x=3\sin x-4\sin^3x,\qquad \cos3x=4\cos^3x-3\cos x. If tan⁡x\tan x and tan⁡3x\tan3x are defined and 1−3tan⁡2x≠01-3\tan^2x\ne0, then tan⁡3x=(3tan⁡x−tan⁡3x)/(1−3tan⁡2x)\tan3x=(3\tan x-\tan^3x)/(1-3\tan^2x). The conventions and prerequisite facts used below are recorded in Double-angle and quadratic power-reduction identities, The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Tangent, cotangent, secant, and cosecant on their exact natural domains.

Facts & Assumptions

Given: A real xx satisfying the displayed tangent hypotheses where required.

Proof

technique · direct
1.1

Expand sin⁡(2x+x)\sin(2x+x) and cos⁡(2x+x)\cos(2x+x) by addition, then substitute the double-angle identities.

given
1.2

Use sin⁡2x=1−cos⁡2x\sin^2x=1-\cos^2x and cos⁡2x=1−sin⁡2x\cos^2x=1-\sin^2x to collect the stated cubic forms.

algebra
2.1

Divide the two cubic identities under the stated nonzero conditions and cancel the common cosine power.

algebra∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources