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Cos A + Cos B

Cos A + Cos B, an important cosine function identity in trigonometry, is used to find the sum of values of cosine function for angles A and B. It is one of the sum to product formulas used to represent the sum of cosine function for angles A and B into their product form. The result for Cos A + Cos B is given as 2 cos ½ (A + B) cos ½ (A - B).

Let us understand the Cos A + Cos B formula and its proof in detail using solved examples.

1.	What is Cos A + Cos B Identity in Trigonometry?
2.	Cos A + Cos B Sum to Product Formula
3.	Proof of Cos A + Cos B Formula
4.	How to Apply Cos A + Cos B Formula?
5.	FAQs on Cos A + Cos B

What is Cos A + Cos B Identity in Trigonometry?

The trigonometric identity Cos A + Cos B is used to represent the sum of the cosine of angles A and B, Cos A + Cos B in the product form using the compound angles (A + B) and (A - B). We will study the Cos A + Cos B formula in detail in the following sections.

Cos A + Cos B Sum to Product Formula

The Cos A + Cos B sum to product formula in trigonometry for angles A and B is given as,

Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B)

Here, A and B are angles, and (A + B) and (A - B) are their compound angles.

expansion of sin A + sin B formula

Proof of Cos A + Cos B Formula

We can give the proof of Cos A + Cos B trigonometric formula using the expansion of cos(A + B) and cos(A - B) formula. As we stated in the previous section, we write Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B).

Let us assume that (α + β) = A and (α - β) = B. We know, using trigonometric identities,

2α = A + B
⇒ α = (A + B)/2

2β = A - B
⇒ β = (A - B)/2

½ [cos(α + β) + cos(α - β)] = cos α cos β, for any angles α and β.

[cos(α + β) + cos(α - β)] = 2 cos α cos β

⇒ Cos A + Cos B = 2 cos ½(A + B) cos ½(A - B)

Hence, proved.

How to Apply Cos A + Cos B?

We can apply the Cos A + Cos B formula as a sum to the product identity to make the calculation easier when it is difficult to find the cosine of given angles. Let us understand its application using the example of cos 60º + cos 30º. We will solve the value of the given expression by 2 methods, using the formula and by directly applying the values, and compare the results. Have a look at the below-given steps.

Compare the angles A and B with the given expression, cos 60º + cos 30º. Here, A = 60º, B = 30º.
Solving using the expansion of the formula Cos A + Cos B, given as, Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B), we get,
Cos 60º + Cos 30º = 2 cos ½ (60º + 30º) cos ½ (60º - 30º) = 2 cos 45º cos 15º = 2 (1/√2) ((√3 + 1)/2√2) = (√3 + 1)/2.
Also, we know that cos 60º + cos 30º = (1/2 + √3/2) = (1 + √3)/2.

Hence, the result is verified.

☛ Related Topics on Cos A + Cos B:

Let us have a look at a few examples to understand the concept of cos A + cos B better.

Examples Using Cos A + Cos B Identity

Example 1: Using the values of angles from the trigonometric table, solve the expression: 2 cos 52.5º cos 7.5º

Solution:

We can rewrite the given expression as, 2 cos 52.5º cos 7.5º = 2 cos ½ (105)º cos ½ (15)º

Assuming A + B = 105º, A - B = 15º and solving for A and B, we get, A = 60º and B = 45º.

⇒ 2 cos ½ (105)º cos ½ (15)º = 2 cos ½ (60º + 45º) cos ½ (60º - 45º)

We know, Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B)

2 cos ½ (60º + 45º) cos ½ (60º - 45º) = cos 60º + cos 45º = (1/2) + (1/√2).
Example 2: Find the value of cos 160º + cos 20º.

Solution:

We know, Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B)

Here, A = 160º, B = 20º

cos 160º + cos 20º = 2 cos ½ (160º + 20º) cos ½ (160º - 20º)

= 2 cos 90º cos 70º

= 0 [∵ cos 90º = 0]
Example 3: Using cos A + cos B, prove that (sin A + sin B)(sin A - sin B) = - (cos A + cos B)(cos A - cos B).

Solution:

Let us rearrange the given expression.

(sin A + sin B)(sin A - sin B) = - (cos A + cos B)(cos A - cos B) can be written as, (sin A + sin B)/(cos A + cos B) = -(cos A - cos B)/(sin A - sin B)

Here, L.H.S. = (sin A + sin B)/(cos A + cos B)

= [2 sin ½ (A + B) cos ½ (A - B)]/[2 cos ½ (A + B) cos ½ (A - B)]

= sin ½ (A + B)/cos ½ (A + B)

R.H.S. = -(cos A - cos B)/(sin A - sin B)

-[- 2 sin ½ (A + B) sin ½ (A - B)]/[2 cos ½ (A + B) sin ½ (A - B)]

= -[- sin ½ (A + B)]/[cos ½ (A + B)]

= sin ½ (A + B)/cos ½ (A + B)

⇒ L.H.S. = R.H.S.

Hence, proved.
Example 4: Verify the given expression using expansion of Cos A + Cos B: cos 70º + sin 70º = √2 cos 25º

Solution:

We have, L.H.S. = cos 70º + sin 70º

Since, sin 70º = sin(90º - 20º) = cos 20º

⇒ cos 70º + sin 70º = ⇒ cos 70º + cos 20º

Using Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B)

⇒ cos 70º + cos 20º = 2 cos ½ (70º + 20º) cos ½ (70º - 20º)

= 2 cos 45º cos 25º

= √2 cos 25º

= R.H.S.

Hence, verified.

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Practice Questions on Cos A + Cos B

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FAQs on Cos A + Cos B

What is Cos A + Cos B in Trigonometry?

Cos A + Cos B is an identity or trigonometric formula, used in representing the sum of cosine of angles A and B, Cos A + Cos B in the product form using the compound angles (A + B) and (A - B). Here, A and B are angles.

What is the Formula of Cos A + Cos B?

Cos A + Cos B formula, for two angles A and B, can be given as, Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B). Here, (A + B) and (A - B) are compound angles.

What is the Expansion of Cos A + Cos B in Trigonometry?

The expansion of Cos A + Cos B formula is given as, Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B), where A and B are any given angles.

How to Prove the Expansion of Cos A + Cos B Formula?

The expansion of Cos A + Cos B, given as Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B), can be proved using the 2 cos α cos β product identity in trigonometry. Click here to check the detailed proof of the formula.

How to Use Cos A + Cos B Formula?

To use Cos A + Cos B formula in a given expression, compare the expansion, Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B) with given expression and substitute the values of angles A and B.

What is the Application of Cos A + Cos B Formula?

Cos A + Cos B formula can be applied to represent the sum of cosine of angles A and B in the product form of cosine of (A + B) and cosine of (A - B), using the formula, Cos A + Cos B = 2 cos ½ (A + B) cos ½ (A - B).

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