Sum difference product

sinθ+sinφ= 2 sin[(θ+φ)/2]cos[(θ-φ)/2]

sinθ-sinφ= 2 cos[(θ+φ)/2]sin[(θ-φ)/2]

cosθ+cosφ= 2 cos[(θ+φ)/2]cos[(θ-φ)/2]

cosθ-cosφ=-2 sin[(θ+φ)/2]sin[(θ-φ)/2]

tanA+tanB = sin(A+B)/cosa cosb = tan(A+B)( 1-tanA tanB)

tanA-tanB = sin(A-B)/cosa cosb = tan(A-B)( 1+tanA tanB)

Sum and difference of products

sinαsinβ=-[cos(α+β)-cos(α-β)]/2

cosαcosβ = [cos(α+β)+cos(α-β)]/2

sinαcosβ = [sin(α+β)+sin(α-β)]/2

cosαsinβ = [sin(α+β)-sin(α-β)]/2

Other commonly used trigonometric formulas

(sinα)^2-(sinβ)^2=(cosβ)^2-(cosα)^2=sin(α+β)sin(α-β)

(cosα)^2-(sinβ)^2=(cosβ)^2-(sinα)^2=cos(α+β)cos(α-β)

In triangle ABC,

cotAcotB+cotBcotC+cotAcotC = tan(A/2)tan(B/2)+tan(B/2)tan(C/2)+tan(C/2)tan(A/2)= 1

Solid geometry, Steiner theorem: in tetrahedron ABCD, let the angle formed by a straight line AB and CD be (AB, CD) and the distance be d(AB, CD), then

cos(ab,cd)=|[(ac^2+bd^2)-(ad^2+bc^2)]/2ab*cd|

d(AB，CD)=6V/[AB*CD*sin(AB，CD)]

De Morgan formula in set

Cu(A∪B)=(CuA)∩(CuB)

Cu(A∩B)=(CuA)∩(CuB)

Common formulas of complex numbers

Z = r(cosθ+is innθ), then z n = r n (cos θ+isinnθ).

Other commonly used formulas, such as plane geometry, number theory and combination, are too biased towards competition, so I won't talk about it.

This is all done by hand. . . I'm exhausted. Give me points ~ ~