Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedprecheck 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.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The addition formulas for sine and cosine

Statement

For all real x,y, sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y, cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y.

Facts & Assumptions

Given: A fixed real y and variable real x.

[L1]

The harmonic-oscillator initial-value problem has the unique solution stated in Existence and uniqueness for y''=-y with prescribed initial data.

[L2]

Sine and cosine have the derivatives and values at zero of The derivatives of sine and cosine are cosine and minus sine, and the chain rule applies (The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f∘g is differentiable at c with (f∘g)′(c)=f′(g(c)) g′(c)).

Proof

technique · direct
1.1L2

The function x↦sin⁡(x+y) solves u′′=−u with initial values u(0)=sin⁡y, u′(0)=cos⁡y.

1.2L2algebra

The function x↦sin⁡xcos⁡y+cos⁡xsin⁡y has the same equation and initial values.

2.1step 1.1step 1.2L1

Uniqueness in [L1] gives the sine addition formula.

3.1L1L2∎

Repeating the argument for x↦cos⁡(x+y) and cos⁡xcos⁡y−sin⁡xsin⁡y gives the cosine addition formula.

Depends on

Dependency tree · 2 levels

Sources