Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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Pythagorean and parity identities for all six trigonometric functions on their natural domains

Statement

For every real xx, sin⁡2x+cos⁡2x=1\sin ^2x+\cos ^2x=1, sin⁡(−x)=−sin⁡x\sin(-x)=-\sin x, and cos⁡(−x)=cos⁡x\cos(-x)=\cos x. Wherever the displayed quotients are defined, tan⁡(−x)=−tan⁡x\tan(-x)=-\tan x, cot⁡(−x)=−cot⁡x\cot(-x)=-\cot x, sec⁡(−x)=sec⁡x\sec(-x)=\sec x, csc⁡(−x)=−csc⁡x\csc(-x)=-\csc x, 1+tan⁡2x=sec⁡2x1+\tan^2x=\sec^2x, and 1+cot⁡2x=csc⁡2x1+\cot^2x=\csc^2x. The conventions and prerequisite facts used below are recorded in 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 and the quotient definitions on their stated nonzero-denominator domains.

Proof

technique · direct
1.1

The sine--cosine Pythagorean and parity identities give the first three equalities.

given
1.2

On the domain of tangent, divide sin⁡2x+cos⁡2x=1\sin ^2x+\cos ^2x=1 by cos⁡2x\cos ^2x; on the domain of cotangent divide it by sin⁡2x\sin ^2x.

algebra
2.1

Applying the quotient definitions to the parity equalities gives the four quotient parity laws without introducing a zero denominator.

given∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 18 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