## Formula and Derivation of 2 Cos A Cos B

2 Cos A Cos B is the product to sum trigonometric formulas that are used to rewrite the product of cosines into sum or difference. The 2 cos A cos B formula can help solve integration formulas involving the product of trigonometric ratio such as cosine. The formula of 2 Cos A Cos B can also be very helpful in simplifying the trigonometric expression by considering the product term such as Cos A Cos B and converting it into sum.

Here, we will look at the 2 Cos A Cos B formula and how to derive the formula of 2 Cos A Cos B.

### 2 Cos A Cos B Formula Derivation

The 2 Cos A Cos B formula can be derived by observing the sum and difference formula for cosine.

As we know,

Cos ( A + B ) = Cos A Cos B - Sin A Sin B... (1)

Cos ( A- B ) = Cos A Cos B + Sin A Sin B…….(2)

Adding the equation (1) and (2), we get

Cos (A + B) + Cos (A - B) = Cos A Cos B - Sin A Sin B + Cos A Cos B + Sin A Sin B

Cos (A + B) + Cos (A - B) = 2 Cos A Cos B (The term Sin A Sin B is cancelled due to the opposite sign).

Therefore, the formula of 2 Cos A Cos B is given as:

In the above 2 Cos A Cos B formula, the left-hand side is the product of cosine whereas the right-hand side is the sum of the cosine.

### 2 Cos A Cos B Formula Application

1. Express 2 Cos 7x Cos 3y as a Sum

Solution:

Let A = 7x and B = 3y

Using the formula:

2 Cos A Cos B = Cos (A + B) + Cos (A - B)

Substituting the values of A and B in the above formula, we get

2 Cos A Cos B = Cos (7x + 3y) + Cos (7x - 3y)

2 Cos A Cos B = Cos 10x + Cos 4y

Hence, 2 Cos 7x Cos 3y = Cos 10x + Cos 4y

### Conclusion

This article discusses 2 Cos A Cos B formulas, The Cos A Cos B is a formula that is derived using the sum and difference trigonometric identity for cosine. The formula is widely used in solving integration problems.

**1. What is a Cosine Function?**

The cosine function is one of the most important functions in trigonometry. In a right-angled triangle, the cosine function is defined as the ratio of the length of the base to the length of the hypotenuse side.

One of the most basic mathematical trigonometric functions is the sine function. The cosine function, which varies as the angle fluctuates, is the ratio of the lengths of the side nearest to the angle and the hypotenuse of a right-angled triangle. For acute angles, it is defined in the context of a right-angled triangle.

Working of cosine function:

COS Function Summary in Excel. The cosine of an angle in radians is calculated by the COS function in excel.

Calculate the cosine of a given angle in radians.

The value of the cosine.

The angle in radians for which you desire the cosine is = COS (number).

The cosine of a given angle in radians is calculated by the COS function.

**2. What is the Use of the Cos A Cos B Formula?**

The Cos A Cos B formula is used in solving the integration problems. Cos A + Cos B in the product form utilizing the compound angles (A + B) and the trigonometric identity Cos A + Cos B is used to indicate the addition of the cosine of angles A and B. (A - B).

**3. How to Apply Cos A + Cos B?**

We can use the Cos A + Cos B formula as a sum to the product identity to simplify the calculation when obtaining the cosine of given angles is difficult. To demonstrate how it works, consider the example of cos60^{o}+cos30^{o}. We'll compare the outcomes of two methods for determining the value of a given expression: using the formula and applying the values right away. Examine the actions outlined below.

Compare the angles A and B using the provided formula, cos 60^{o} + cos 30^{o}.

In this situation, A = 60^{o}, B = 30^{o}

Expansion of the formula

Cos A + Cos B, given as, Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B),

Cos 60^{o} + Cos 30^{o} = 2 cos ½ (60^{o} + 30^{o}) cos ½ (60^{o} - 30^{o}) = 2 cos 45^{o} cos 15^{o} = 2 (1/√2) ((√3 + 1)/2√2) = (√3 + 1)/2.