TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-02

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Triple-angle identities for sine, cosine, and tangent

Statement

For every real $x$ , $\sin 3 x = 3 \sin x - 4 \sin^{3} x, \cos 3 x = 4 \cos^{3} x - 3 \cos x .$ If $\tan x$ and $\tan 3 x$ are defined and $1 - 3 \tan^{2} x \neq 0$ , then $\tan 3 x = (3 \tan x - \tan^{3} x) / (1 - 3 \tan^{2} x)$ . 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 $x$ satisfying the displayed tangent hypotheses where required.

Proof

technique · direct

1.1

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

given

1.2

Use $\sin^{2} x = 1 - \cos^{2} x$ and $\cos^{2} x = 1 - \sin^{2} x$ 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

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)