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Pythagorean and parity identities for all six trigonometric functions on their natural domains
Statement
For every real , , , and . Wherever the displayed quotients are defined, , , , , , and . 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 and the quotient definitions on their stated nonzero-denominator domains.
Proof
The sine--cosine Pythagorean and parity identities give the first three equalities.
On the domain of tangent, divide by ; on the domain of cotangent divide it by .
Applying the quotient definitions to the parity equalities gives the four quotient parity laws without introducing a zero denominator.
Depends on
Used by
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 Corollary
- Principal arcsine has no finite derivative at -1 or 1 Counterexample
- Machin's formula π/4=4 arctan(1/5)-arctan(1/239) Example
- The arc length of one sine period is 4√2 E(1/√2) Example
- Zero mean curvature implies a submanifold is totally geodesic False statement
- Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
- Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions Theorem
- Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series Theorem
Dependency tree · two levels
8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)