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

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Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions

Statement

For every real $x$ , $\sin (π / 2 - x) = \cos x, \cos (π / 2 - x) = \sin x, \sin (π - x) = \sin x, \cos (π - x) = - \cos x,$ and the quarter-turn and reflection formulas are $\sin (x + π / 2) = \cos x, \cos (x + π / 2) = - \sin x, \sin (- x) = - \sin x, \cos (- x) = \cos x .$ On the common natural domains of the two sides, $\tan (π / 2 - x) = \cot x, \cot (π / 2 - x) = \tan x, \sec (π / 2 - x) = \csc x, \csc (π / 2 - x) = \sec x,$ and $\tan (π - x) = - \tan x, \cot (π - x) = - \cot x, \sec (π - x) = - \sec x, \csc (π - x) = \csc x .$ The conventions and prerequisite facts used below are recorded in Pythagorean and parity identities for all six trigonometric functions on their natural domains, Quarter-turn values and shifts by pi/2 and pi, The addition formulas for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Tangent, cotangent, secant, and cosecant on their exact natural domains.

Facts & Assumptions

Given: A real $x$ .

[L1]

Quarter-turn values and shifts by pi/2 and pi states that, for every real $x$ , $\sin (x + π / 2) = \cos x$ and $\cos (x + π / 2) = - \sin x$ .

[L2]

The addition formulas for sine and cosine states that, for all real $a, b$ , $\sin (a + b) = \sin a \cos b + \cos a \sin b$ and $\cos (a + b) = \cos a \cos b - \sin a \sin b$ .

[L3]

Pythagorean and parity identities for all six trigonometric functions on their natural domains gives $\sin (- x) = - \sin x$ and $\cos (- x) = \cos x$ .

[L4]

Tangent, cotangent, secant, and cosecant on their exact natural domains defines $\tan t = \sin t / \cos t$ , $\cot t = \cos t / \sin t$ , $\sec t = 1 / \cos t$ , and $\csc t = 1 / \sin t$ on their natural domains.

Proof

technique · direct

1.1

By [L1] with $x$ replaced by $- x$ , and then [L3], $\sin (π / 2 - x) = \cos x$ and $\cos (π / 2 - x) = \sin x$ ; [L1] also gives the displayed quarter-turn formulas.

L1L3

1.2

Apply [L2] to $π + (- x)$ and use the values at $π$ supplied by [L1] (put $x = π / 2$ ) together with [L3]. This gives $\sin (π - x) = \sin x$ and $\cos (π - x) = - \cos x$ .

L1L2L3

2.1

Substitute the cofunction and supplementary sine--cosine equalities into [L4]. The stated nonvanishing conditions are exactly those that make both quotient or reciprocal expressions defined, so this yields all eight displayed identities.

L4step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

17 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)