Cos A  Cos B
Cos A  Cos B, an important identity in trigonometry, is used to find the difference of values of cosine function for angles A and B. It is one of the difference to product formulas used to represent the difference of cosine function for angles A and B into their product form. The result for Cos A  Cos B is given as 2 sin ½ (A + B) sin ½ (B  A).
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 Difference 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 difference of 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 Difference to Product Formula
The Cos A  Cos B difference to product formula in trigonometry for angles A and B is given as,
Cos A  Cos B =  2 sin ½ (A + B) sin ½ (A  B)
or
Cos A  Cos B = 2 sin ½ (A + B) sin ½ (B  A)
Here, A and B are angles, and (A + B) and (A  B) are their compound angles.
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 sin ½ (A + B) sin ½ (B  A).
Let us assume two compound angles A and B, given as A = X + Y and B = X  Y,
⇒ Solving, we get,
X = (A + B)/2 and Y = (A  B)/2
We know, cos(X + Y) = cos X cos Y  sin X sin Y
cos(X  Y) = cos X cos Y + sin X sin Y
cos(X + Y)  cos(X  Y) = 2 sin X sin Y
⇒ Cos A  Cos B =  2 sin ½ (A + B) sin ½ (A  B)
⇒ Cos A  Cos B = 2 sin ½ (A + B) sin ½ (B  A)
Hence, proved.
How to Apply Cos A  Cos B Formula?
We can apply the Cos A  Cos B formula as a difference to the product identity. Let us understand its application using an 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 belowgiven 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 sin ½ (A + B) sin ½ (B  A), we get,
Cos 60º  Cos 30º = 2 sin ½ (60º + 30º) sin ½ (30º  60º) =  2 sin 45º sin 15º =  2 (1/√2) ((√3  1)/2√2) = (1  √3)/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: Find the value of cos 165º  cos 15º.
Solution:
We know, Cos A  Cos B = 2 sin ½ (A + B) sin ½ (B  A)
Here, A = 165º, B = 15º
cos 165º  cos 15º = 2 sin ½ (165º + 15º) sin ½ (165º  15º)
= 2 sin 90º sin 75º
= 2 sin 75º
= 2 sin(45º + 30º) = 2(sin 45º cos 30º + sin30º cos45º)
= 2((1/√2) (√3/2) + (1/2)(1/√2))
= (√3 + 1)/√2

Example 2: Using the values of angles from the trigonometric table, solve the expression: 2 sin 67.5º sin 22.5º
Solution:
We can rewrite the given expression as, 2 sin 67.5º sin 22.5º = 2 sin ½ (135)º sin ½ (45)º
Assuming A + B = 135º, A  B = 45º and solving for A and B, we get, A = 90º and B = 45º.
⇒ 2 sin ½ (135)º sin ½ (45)º = 2 sin ½ (90º + 45º) sin ½ (90º  45º)
We know, Cos A  Cos B = 2 sin ½ (A + B) sin ½ (B  A)
2 sin ½ (90º + 45º) sin ½ (90º  45º) = cos 90º  cos 45º = (1/√2).

Example 3: Solve the given expression, (cos x  cos 5x)/(cos 2x  cos 4x).
Solution:
We have,
(cos x  cos 5x)/(cos 2x  cos 4x) = [2 sin ½ (x + 5x) sin ½ (x  5x)]/[2 sin ½ (2x + 4x) sin ½ (2x  4x)]
= [sin 3x sin(2x)]/[sin 3x sin(x)]
= (sin 3x sin 2x)/(sin 3x sin x)
= sin 2x cosec x

Example 4: Verify the given expression using expansion of Cos A  Cos B: cos 70º  sin 70º = √2 sin 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 sin ½ (A + B) sin ½ (B  A)
⇒ cos 70º  cos 20º = 2 sin ½ (70º + 20º) sin ½ (70º  20º)
= 2 sin 45º sin 25º
= √2 sin 25º
Hence, verified.
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 difference 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.
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 sin ½ (A + B) sin ½ (B  A) with given expression and substitute the values of angles A and B.
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 sin ½ (A + B) sin ½ (B  A). 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 sin ½ (A + B) sin ½ (B  A), 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 sin ½ (A + B) sin ½ (B  A), can be proved using the 2 sin X sin Y product identity in trigonometry. Click here to check the detailed proof of the formula.
What is the Application of Cos A  Cos B Formula?
Cos A  Cos B formula can be applied to represent the difference of cosine of angles A and B in the product form of sine of (A + B) and sine of (A  B), using the formula, Cos A  Cos B = 2 sin ½ (A + B) sin ½ (B  A).
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