Angle addition formulas express trigonometric functions of sums of angles alpha+/-beta in terms of functions of alpha and beta. The fundamental formulas of angle addition in trigonometry are given by

sin(alpha+beta)=sinalphacosbeta+sinbetacosalpha(1)sin(alpha-beta)=sinalphacosbeta-sinbetacosalpha(2)cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta(3)cos(alpha-beta)=cosalphacosbeta+sinalphasinbeta(4)tan(alpha+beta)=(tanalpha+tanbeta)/(1-tanalphatanbeta)(5)tan(alpha-beta)=(tanalpha-tanbeta)/(1+tanalphatanbeta).(6)The first four of these are known as the prosthaphaeresis formulas, or sometimes as Simpson's formulas.

The sine and cosine angle addition identities can be compactly summarized by the matrix equation

[cosalpha sinalpha; -sinalpha cosalpha][cosbeta sinbeta; -sinbeta cosbeta]=[cos(alpha+beta) sin(alpha+beta); -sin(alpha+beta) cos(alpha+beta)]. (7)These formulas can be simply derived using complex exponentials and the Euler formula as follows.

cos(alpha+beta)+isin(alpha+beta)=e^(i(alpha+beta))(8)=e^(ialpha)e^(ibeta)(9)=(cosalpha+isinalpha)(cosbeta+isinbeta)(10)=(cosalphacosbeta-sinalphasinbeta)+i(sinalphacosbeta+cosalphasinbeta).(11)Equating real and imaginary parts then gives (1) and (3), and (2) and (4) follow immediately by substituting -beta for beta.

Taking the ratio of (1) and (3) gives the tangent angle addition formula

tan(alpha+beta)=(sin(alpha+beta))/(cos(alpha+beta))(12)=(sinalphacosbeta+sinbetacosalpha)/(cosalphacosbeta-sinalphasinbeta)(13)=((sinalpha)/(cosalpha)+(sinbeta)/(cosbeta))/(1-(sinalphasinbeta)/(cosalphacosbeta))(14)=(tanalpha+tanbeta)/(1-tanalphatanbeta).(15)The double-angle formulas are

sin(2alpha)=2sinalphacosalpha(16)cos(2alpha)=cos^2alpha-sin^2alpha(17)=2cos^2alpha-1(18)=1-2sin^2alpha(19)tan(2alpha)=(2tanalpha)/(1-tan^2alpha).(20)Multiple-angle formulas are given by

sin(nx)=sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)xsin[1/2(n-k)pi](21)cos(nx)=sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)xcos[1/2(n-k)pi],(22)and can also be written using the recurrence relations

sin(nx)=2sin[(n-1)x]cosx-sin[(n-2)x](23)cos(nx)=2cos[(n-1)x]cosx-cos[(n-2)x](24)tan(nx)=(tan[(n-1)x]+tanx)/(1-tan[(n-1)x]tanx).(25)TrigAnglesWeissteinThe angle addition formulas can also be derived purely algebraically without the use of complex numbers. Consider the small right triangle in the figure above, which gives

a=(sinalpha)/(cos(alpha+beta))(26)b=sinalphatan(alpha+beta).(27)Now, the usual trigonometric definitions applied to the large right triangle give

sin(alpha+beta)=(sinbeta+a)/(cosalpha+b)(28)=(sinbeta+(sinalpha)/(cos(alpha+beta)))/(cosalpha+sinalpha(sin(alpha+beta))/(cos(alpha+beta)))(29)cos(alpha+beta)=(cosbeta)/(cosalpha+b)(30)=(cosbeta)/(cosalpha+sinalpha(sin(alpha+beta))/(cos(alpha+beta))).(31)Solving these two equations simultaneously for the variables sin(alpha+beta) and cos(alpha+beta) then immediately gives

sin(alpha+beta)=(cosalphasinalpha+cosbetasinbeta)/(cosalphacosbeta+sinalphasinbeta)(32)cos(alpha+beta)=(cos^2beta-sin^2alpha)/(cosalphacosbeta+sinalphasinbeta).(33)These can be put into the familiar forms with the aid of the trigonometric identities

(cosalphacosbeta+sinalphasinbeta)(sinalphacosbeta+sinbetacosalpha)=cosbetasinbeta+cosalphasinalpha (34)and

(cosalphacosbeta+sinalphasinbeta)(cosalphacosbeta-sinalphasinbeta)=cos^2alphacos^2beta-sin^2alphasin^2beta(35)=1-sin^2alpha-sin^2beta(36)=cos^2alpha-sin^2beta(37)=cos^2beta-sin^2alpha,(38)which can be verified by direct multiplication. Plugging (◇) into (◇) and (38) into (◇) then gives

sin(alpha+beta)=sinalphacosbeta+sinbetacosalpha(39)cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta,(40)as before.

TrigAdditionSmileyA similar proof due to Smiley and Smiley uses the left figure above to obtain

sinalpha=(sin(alpha+beta))/(cosbeta+(sinbetacosalpha)/(sinalpha)), (41)from which it follows that

sin(alpha+beta)=sinalphacosbeta+sinbetacosalpha. (42)Similarly, from the right figure,

(sinalpha)/(cosalpha)=(cosbeta)/(sinbeta+(cos(alpha+beta))/(sinalpha)), (43)so

cos(alpha+beta)=cosalphacosbeta-sinalphasinbeta. (44)TrigSubtractionSmileySimilar diagrams can be used to prove the angle subtraction formulas (Smiley 1999, Smiley and Smiley). In the figure at left,

h=(cosalpha)/(cosbeta)(45)x=hsin(alpha-beta)(46)=(sinalpha-hsinbeta)cosalpha,(47)giving

sin(alpha-beta)=sinalphacosbeta-cosalphasinbeta. (48)Similarly, in the figure at right,

h=(cosalpha)/(sinbeta)(49)x=hcos(alpha-beta)(50)=(sinalpha+hcosbeta)cosalpha,(51)giving

cos(alpha-beta)=cosalphacosbeta+sinalphasinbeta. (52)TanSubtractionRenA more complex diagram can be used to obtain a proof from the tan(alpha-beta) identity (Ren 1999). In the above figure, let BF/BE=AD/DE. Then

tan(alpha-beta)=(DE)/(BE)=(AD)/(BF)=(tanalpha-tanbeta)/(1+tanalphatanbeta). (53)An interesting identity relating the sum and difference tangent formulas is given by

(tan(alpha-beta))/(tan(alpha+beta))=(sin(alpha-beta)cos(alpha+beta))/(cos(alpha-beta)sin(alpha+beta))(54)=((sinalphacosbeta-sinbetacosalpha)(cosalphacosbeta-sinalphasinbeta))/((cosalphacosbeta+sinalphasinbeta)(sinalphacosbeta+sinbetacosalpha))(55)=(sinalphacosalpha-sinbetacosbeta)/(sinalphacosalpha+sinbetacosbeta).