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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Quarter-turn values and shifts by pi/2 and pi

Statement

For every real $x$ , $\sin (x + π / 2) = \cos x, \cos (x + π / 2) = - \sin x, \sin (x + π) = - \sin x, \cos (x + π) = - \cos x .$ In particular, $\sin (π / 2) = 1, \cos (π / 2) = 0, \sin π = 0, \cos π = - 1.$

Facts & Assumptions

Given: A real $x$ and $γ = π / 2$ .

[L1]

$\cos γ = 0$ and $\sin γ > 0$ (Pi as twice the smallest positive zero of cosine, Cosine has a smallest positive zero, lying strictly between zero and two, Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3).

[L2]

The addition formulas and Pythagorean identity hold (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine).

Proof

technique · direct

1.1L1L2

From $\cos^{2} γ + \sin^{2} γ = 1$ , $\cos γ = 0$ , and $\sin γ > 0$ , one gets $\sin γ = 1$ .

2.1step 1.1L2

Substituting $γ$ into the addition formulas gives $\sin (x + γ) = \cos x$ and $\cos (x + γ) = - \sin x$ .

3.1step 2.1L1∎

Applying step 2.1 twice gives the shifts by $2 γ = π$ and the listed special values.

Quarter-turn values and shifts by pi/2 and pi

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources