Fundamental Trigonometric Identities
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"arc" Identities
arctanθ=tan−1θ
arcsinθ=sin−1θ
arccosθ=cos−1θ
Quotient and reciprocal identities
tanθ=sinθcosθ
cotθ=cosθsinθ=cscθsecθ=1tanθ
secθ=1cosθ
cscθ=1sinθ
Cofunction Function identities
sinθ=cos(π2−θ)
cosθ=sin(π2−θ)
Even/Odd Functions
cos(−θ)=cos(θ)
sin(−θ)=−sin(θ)
Pythagorean identities
sin2θ+cos2θ=1
tan2θ+1=sec2θ
1+cot2θ=csc2θ
Angle sum and difference identities
sin(α+β)=sinαcosβ+sinβcosα
sin(α−β)=sinαcosβ−sinβcosα
cos(α+β)=cosαcosβ−sinαsinβ
cos(α−β)=cosαcosβ+sinαsinβ
tan(α+β)=tanα+tanβ1−tanαtanβ
tan(α−β)=tanα−tanβ1+tanαtanβ
Double-angle identities
sin2θ=2sinθcosθ
cos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ
tan2θ=2tanθ1−tan2θ
Half-angle identitie
ssinθ2=±√1−cosθ2
cosθ2=±√1+cosθ2
tanθ2=±√1−cosθ1+cosθ=sinθ1+cosθ=1−cosθsinθ
Reduction formulas
sin2θ=1−cos2θ2
cos2θ=1+cos2θ2
tan2θ=1−cos2θ1+cos2θ=sin2θ1+cos2θ=1−cos2θsin2θ