1, sine, cosine and tangent formulas of sum and difference of two angles and sine, cosine and tangent formulas of two angles: cos (α+β) = cos α cos β-sin α sin β cos (α-β) = cos α cos β+sin α sin β.

sin(α+β)= sinαcosβ+cosαsinβsin(α-β)= sinαcosβ-cosαsinβ

tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)

sin 2α= 2 sinαcosαcos 2α=(cosα)^2-(sinα)^2=2(cosα)^2 - 1= 1-2(sinα)^2

Tan2α = 2tanα/[1-(tanα) 2] 2, sine multiple angle formula:

sin2α = 2cosαsinα

Deduction: SIN2A = SIN (a+a) = SINA COSA+COSA SINA = 2 SINA COSA.

Extended formula: sin2a = 2sinacosa = 2tanacosa2 = 2tana/[1+tana2]

Cosine double angle formula:

& lt/B& gt； The cosine double angle formula has three sets of expressions, which are equivalent:

1.cos2a=cosa^2-sina^2=[ 1-tana^2]/[ 1+tana^2]

2.Cos2a= 1-2Sina^2

3.Cos2a=2Cosa^2- 1

Deduction: COS2A = COS (a+a) = COSA COSA-SINA SINA = (COSA) 2-(SINA) 2 = 2 (COSA) 2-1

= 1-2(sinA)^2

Tangent dihedral formula:

& lt/B& gt； tan2α=2tanα/[ 1-(tanα)^2]

Deduction: tan2a = tan (a+a) = (tana+tana)/(1-tanatana) = 2tana/[1-(tana) 2]

3. Auxiliary angle formula: For the function of type acosx+bsinx, we can deform Acosx+BSINX = SQRT (a2+B2) (Acosx/SQRT (A2+B2)+BSINX/SQRT (A2+B2)) and make a point (A2+B2).

∴acosx+bsinx=sqrt(a^2+b^2)sin(x+arctan(a/b))

This is the auxiliary angle formula.

Let the formula to be proved be acosa+BSINA = √ (A2+B2) SIN (A+M) (TANM = A/B).

The following is the proof process:

Let acosA+bsinA=xsin(A+M)

∴acosa+bsina=x((a/x)cosa+(b/x)sina)

Problem, (a/x) 2+(b/x) 2 = 1, SINM = a/x, COSM = B/X.

∴x=√(a^2+b^2)

∴acosA+bsinA=√(a^2+b^2)sin(A+M)，tanM=sinM/cosM=a/b