TheoremStatement: AI-adaptedProof: AI-adaptedprecheck 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 $x$ , $\sin^{2} x + \cos^{2} x = 1$ , $\sin (- x) = - \sin x$ , and $\cos (- x) = \cos x$ . Wherever the displayed quotients are defined, $\tan (- x) = - \tan x$ , $\cot (- x) = - \cot x$ , $\sec (- x) = \sec x$ , $\csc (- x) = - \csc x$ , $1 + \tan^{2} x = \sec^{2} x$ , and $1 + \cot^{2} x = \csc^{2} x$ . 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 $x$ and the quotient definitions on their stated nonzero-denominator domains.

Proof

technique · direct

1.1given

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

1.2algebra

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

2.1given∎

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

Depends on

Dependency tree · 2 levels

Sources

NIST Digital Library of Mathematical Functions, Chapter 4