Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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The subtraction formulas for sine and cosine

Statement

For all real u,vu,v, sin⁡(u−v)=sin⁡ucos⁡v−cos⁡usin⁡v,cos⁡(u−v)=cos⁡ucos⁡v+sin⁡usin⁡v.\sin(u-v)=\sin u\cos v-\cos u\sin v,\qquad \cos(u-v)=\cos u\cos v+\sin u\sin v. The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine.

Facts & Assumptions

Given: Reals u,vu,v.

Proof

technique · direct
1.1

Apply the addition formulas to u+(−v)u+(-v).

given
2.1

Replace sin⁡(−v)\sin(-v) and cos⁡(−v)\cos(-v) by their parity values and simplify.

algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 14 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