How do you derive the product of the sum difference and the product of the difference? Forgot

Accumulation and Difference Formula:

<>1) Proof: <>

<>2) Proof: <>

<>3) Proof: <>

<>4) Proof: <>

High school textbooks do not directly write formulas for the sum and difference and the difference product, but only give practice questions after class to ask you to prove these formulas.

The proof is simple, just take the right side of the equation apart with the formula of the sum of the two corners.

sinαsinβ=-1/2[cos(α+cos(α-

1/2[（cosαcosβ-sinαsinβ)-cosαcosβ+sinαsinβ)]

1/2[-2sinαsinβ]

The others are also the same proof method:

cosαcosβ= 1/2[cos(α+cos(α-

sinαcosβ= 1/2[sin(α+sin(α-

cosαsinβ= 1/2[sin(α+sin(α-

sinθ+sinφ=2sin(θ/2+θ/2)cos(θ/2-φ/2)

2[sinθ/2cosφ/2+cosθ/2sinφ/2][cosθ/2cosφ/2+

sinφ/2sinθ/2]

2cosθ/2sinθ/2+2sinφ/2cosφ/2

sinθ+sinφ

The others are proved in the same way:

sinθ-sinφ=2cos(θ/2+φ/2)sin(θ/2-φ/2)

cosθ+cosφ=2cos(θ/2+φ/2)cos(θ/2-φ/2)

cosθ-cosφ=-2sin(θ/2+φ/2)sin(θ/2-φ/2)

It is not difficult to see that the sum and difference products are deduced from the product sum and difference formula.

I just know, haha, let's see if you need it

1. Formula for Accumulation and Difference:

sinαsinβ=-[cos(α+cos(α-

cosαcosβ=[cos(α+cos(α-

sinαcosβ=[sin(α+sin(α-

cosαsinβ=[sin(α+sin(α-

The formula of product sum difference is derived from the sum angle formula of sine or cosine and the difference angle formula by addition and subtraction. The latter two formulas can be combined into one:

sinαcosβ=[sin(α+sin(α-

2. Sum difference product formula.

sinθ+sinφ=2sincos

sinθ-sinφ=2cossin

cosθ+cosφ=2coscos

cosθ-cosφ=-2sinsin

The sum and difference product formula is the inverse form of the product and difference formula, and it should be noted that:

The first two formulas can be combined into one: sin + sin = 2sincos

The idea of "solving the system of equations" is used in the derivation of the formula of the product sum and the idea of "commutation" is used in the derivation of the formula of the difference product.

Only the sum and difference of the function with the same name with the same absolute value of the coefficient can be directly formulated into the product, if the sum or difference of a sine and a cosine is used, the function of the same name should be formulated with the induction formula first, and then the product of the formula should be used.

The unity deformation is also a kind of sum difference product.

The sum and difference product of trigonometric functions can be understood as factorization in algebra, so what role does factorization play in algebra, and what role does the sum product formula play in trigonomy.

Formula for Accumulation and Inclusion Difference:

<>1) Proof: <>

<>2) Proof: <>

<>3) Proof: Finch eggplant erection.

<>4) Proof: <>

Derivation process: It can be derived by the formula of the sum difference of the product, or it can be obtained by the sum angle formula, which is proved by the sum angle formula below.

By and the angle formula has:

<> addition and subtraction of the two formulas can be obtained or to the above formula, and the same can be proved by the formula.

For (5), (6), there are:

Certification. <>

Methods of memorization. 1. Memorize only two formulas or even one.

You can just memorize the first and third of the four formulas above.

The second formula is the <> of the brothers

i.e. <>

This can be done using the first formula. Similarly, in the fourth formula, <

This can be solved with a third formula.

If you are familiar enough with the induction formula, you can convert all the cosines to sine during the operation, so that you can only remember the first formula. I can afford one or two when I use it.

2. Multiply the blind fruit by 2

The easiest way to remember this is to use the range of trigonometric functions. Both the sine and cosine ranges [-1,1], and the range of their products should also be [-1,1], while the range of sum and difference is [-2,2], so multiplying by 2 is necessary.

It can also be remembered by its proof, because after the formula of the sum of the two angles, the two items that are not canceled are the same and cause a coefficient of 2, such as:

So you need to multiply by 2 at the end.