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

Statement

For every real xx, sin⁡(π/2−x)=cos⁡x,cos⁡(π/2−x)=sin⁡x,sin⁡(π−x)=sin⁡x,cos⁡(π−x)=−cos⁡x,\sin(\pi/2-x)=\cos x,\quad \cos(\pi/2-x)=\sin x,\quad \sin(\pi-x)=\sin x,\quad \cos(\pi-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.\sin(x+\pi/2)=\cos x,\quad \cos(x+\pi/2)=-\sin x,\quad \sin(-x)=-\sin x,\quad \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,\tan(\pi/2-x)=\cot x,\quad \cot(\pi/2-x)=\tan x,\quad \sec(\pi/2-x)=\csc x,\quad \csc(\pi/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.\tan(\pi-x)=-\tan x,\quad \cot(\pi-x)=-\cot x,\quad \sec(\pi-x)=-\sec x,\quad \csc(\pi-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 xx.

[L1]

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

[L2]

The addition formulas for sine and cosine states that, for all real a,ba,b, sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b\sin(a+b)=\sin a\cos b+\cos a\sin b and cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\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\sin(-x)=-\sin x and cos⁡(−x)=cos⁡x\cos(-x)=\cos x.

[L4]

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

Proof

technique · direct
1.1

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

L1L3
1.2

Apply [L2] to π+(−x)\pi+(-x) and use the values at π\pi supplied by [L1] (put x=π/2x=\pi/2) together with [L3]. This gives sin⁡(π−x)=sin⁡x\sin(\pi-x)=\sin x and cos⁡(π−x)=−cos⁡x\cos(\pi-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 · next 3 levels

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Sources