Trigonometric ratios are defined using the proportions of the sides of a right-angled triangle. The six fundamental trigonometric ratios are \(\sin\), \(\cos\), \(\tan\), \(\cot\), \(\sec\), and \(cosec\). Each of these ratios has a separate formula. It uses the three sides and three angles of a right-angled triangle. In order to rewrite the product of cosines as a sum or difference, the trigonometric formula for \(\cos(a)\cos(b)\) is used.

Integration problems regarding the product of a trigonometric ratio, such as cosine, can be solved using the cos a cos b formula. By taking into account the product term, such as cos a cos b, and transforming it into sum, the cos a cos b formula proves to be quite helpful in reducing the trigonometric statement.

What is Cos a Cos b in Trigonometry?

Standard identities are algebraic expressions that help simplify mathematical problems. An identity is an equation that is true no matter what values are used for its variables.

One commonly used trigonometric identity is: cos(a) cos(b) = (1/2) [cos(a + b) + cos(a − b)]

This formula is useful for solving problems that involve cosine or other trigonometric products, such as when integrating functions.

In trigonometry, this is called a defactorization formula. A product is “defactored” when it is rewritten as a sum or difference, making calculations easier.

Cos A – Cos B: Difference to Product Formula

In trigonometry, there is a useful formula that helps us change the difference of two cosine terms into a product of sine terms. This is known as the difference to product formula for cosine.

The formula is written as:

cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)

You can also write it in another form as:

cos A − cos B = 2 sin((A + B)/2) sin((B − A)/2)

Both forms mean the same thing because sin((B − A)/2) = −sin((A − B)/2).

Example:

Let’s take an example: Find the value of cos 70° − cos 50°.

Using the formula:

cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)

Substitute the values:

cos 70° − cos 50° = −2 sin((70 + 50)/2) sin((70 − 50)/2)
= −2 sin(60°) sin(10°)

So,

cos 70° − cos 50° = −2 × (√3/2) × sin(10°)

This gives the simplified form using the difference to product rule.

Proof of Cos A - Cos B Formula

We know that,

\(\cos{\theta} = \frac{\text{adjacent Side}}{\text{Hypotenuse}}\)

\(\sin{\theta} = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\)

We know that

\(\cos(a + b) = \cos a \cos b – \sin a \sin b ………….. (i)\)

\(\cos(a – b) = \cos a \cos b + \sin a \sin b ………….. (ii)\)

When we combine the above-given equations (i) and (ii), we get

\(\cos (a + b) + \cos (a – b) = \cos a \cos b – \sin a \sin b + \cos a \cos b + \sin a \sin b\)

\(\cos (a + b) + \cos (a – b) = 2 \cos a \cos b\)

Thus,

\(2 \cos a \cos b = \cos (a + b) + \cos (a – b)\)

\(\cos a \cos b=(\frac{1}{2})[\cos (a + b) + \cos (a – b)]\)

Learn more about Line Graph here.

How to Apply Cos a Cos b Formula?

Knowing the cos a cos b formula will help us understand how to use it to address a variety of problems. Simple trigonometric problems and challenging integration problems can both be solved using this identity. To apply cos a cos b identity, we can learn the steps provided below.

Let’s take an example problem.

Example: Integrate: \({\int}cos8x(1 + cos2x)\)

Now we know that we can apply the u.v rule of integration to solve this problem, but it will be very complicated and can cause errors. Instead, we will work towards simplifying the expression.

Step 1: Simplifying the expression.

\(cos8x(1 + cos2x)\)

\({\implies}cos8x + cos8x cos2x\)

Now we still have two cos terms in multiplication, we can simplify it further by using the formula we just learned.

Step 2: Applying the cos a cos b identity.

\(cos8x cos2x\)

Here \(cos a = cos8x {\implies} a = 8x\) and \(cos b = cos 2x {\implies} b = 2x\)

We know that,

\(cos a cos b=(\frac{1}{2})[cos (a + b) + cos (a – b)]\)

Thus,

\(cos8x cos2x = (\frac{1}{2})[cos (8x + 2x) + cos (8x – 2x)]\)

\({\implies}cos8x cos2x = (\frac{1}{2})cos (10x) + cos (6x)\)

Thus, the equation becomes,

\({\implies}cos8x + (\frac{1}{2})[cos10x + cos6x]\)

\({\implies}cos8x + (\frac{1}{2})cos10x + (\frac{1}{2})cos6x\)

Now we can easily integrate this expression.

Step 3: Solve the Problem

\({\int}[cos8x + (\frac{1}{2})cos10x + (\frac{1}{2})cos6x]\)

\({\implies} {\int}cos8x + {\int}(\frac{1}{2})cos10x + {\int}(\frac{1}{2})cos6x\)

\({\implies} 8sin8x +(\frac{10}{2})sin10x + (\frac{6}{2})sin6x + C\)

\({\implies} 8sin8x + 5sin10x + 3sin6x + C\)

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Properties of Cos A Cos B 

In trigonometry, the product of two cosine terms, written as cos A × cos B, has some important properties. These properties make it easier to simplify expressions, solve equations, and work with trigonometric functions.

1. Product-to-Sum Formula

The most important property is that the product of two cosines can be written as a sum of cosines:

cos A cos B = (1/2) [cos(A + B) + cos(A − B)]

This formula converts the multiplication of two cosines into addition. It is useful for simplifying expressions and integrating trigonometric functions.

2. Commutative Property

cos A cos B = cos B cos A

The order of multiplication does not matter. Multiplying cos A by cos B gives the same result as multiplying cos B by cos A.

3. Symmetry with Negative Angles

cos(−A) cos B = cos A cos B

Cosine is an even function, so changing the sign of one angle does not change the product.

4. Zero Property

cos A cos B = 0 if either cos A = 0 or cos B = 0

If either of the cosine terms is zero, the whole product becomes zero.

5. Range of Values

The value of cos A cos B always lies between −1 and 1. This is because each cosine value is between −1 and 1, and the product of two numbers in this range also lies in [−1, 1].

6. Useful in Integration and Solving Equations

Using the product-to-sum formula, cos A cos B can be simplified when integrating:

∫ cos A cos B dx = (1/2) ∫ [cos(A + B) + cos(A − B)] dx

Similarly, it helps in solving trigonometric equations by turning products into sums.

Cos a Cos b Solved Examples

Here are some solved examples on cos a cos b.

Solved Example 1: Prove That: \(cos 2x cos (\frac{x}{2}) – cos 3x cos (\frac{9x}{2}) = sin 5x sin (\frac{5x}{2})\)

Solution:

\(LHS = cos 2x cos (\frac{x}{2}) – cos 3x cos (\frac{9x}{2})\)

We know that,

\(cos a cos b = (\frac{1}{2})[cos (a + b) + cos (a – b)]\)

\({\implies} cos 2x cos (\frac{x}{2})\)

Here \(cos a = cos2x {\implies} a = 2x\) and \(cos b = cos (\frac{x}{2}) {\implies} b = (\frac{x}{2})\)

\({\implies} cos 2x cos (\frac{x}{2}) = (\frac{1}{2})[cos(\frac{5x}{2}) + cos(\frac{3x}{2})]\)

Similarly,

\({\implies} cos 3x cos (\frac{9x}{2})\)

Here \(cos a = cos3x {\implies} a = 3x\) and \(cos b = cos (\frac{9x}{2}) {\implies} b = (\frac{9x}{2})\)

\({\implies} cos 3x cos (\frac{9x}{2}) = (\frac{1}{2})[cos(\frac{15x}{2}) + cos(\frac{-3x}{2})]\)

\({\implies} cos 3x cos (\frac{9x}{2}) = (\frac{1}{2})[cos(\frac{15x}{2}) – cos(\frac{3x}{2})] …………….. [cos(-x)=-cosx]\)

Thus,

\(LHS = (\frac{1}{2})[cos(\frac{5x}{2}) + cos(\frac{3x}{2})] + (\frac{1}{2})[cos(\frac{15x}{2}) – cos(\frac{3x}{2})]\)

\(LHS = (\frac{1}{2})[cos(\frac{5x}{2}) + cos(\frac{3x}{2}) + cos(\frac{15x}{2}) – cos(\frac{3x}{2})]\)

\(LHS = (\frac{1}{2})[cos(\frac{5x}{2}) + cos(\frac{15x}{2})]\)

We know that, \(cos x – cos y = – 2 sin \frac{(x+y)}{2} sin \frac{(x -y)}{2}\)

Thus,

\(LHS = \frac{1}{2} [- 2 sin\frac{((\frac{5x}{2}) + (\frac{15x}{2}))}{2} sin \frac{((\frac{5x}{2}) – (\frac{15x}{2}))}{2}]\)

\(LHS = – sin 5x sin(\frac{-5x}{2})\)

\(LHS = sin 5x sin(\frac{5x}{2})\)

= RHS

LHS = RHS

Hence, Proved.

Solved Example 2: Integrate 8 cos y cos 2y

Solution:

We are asked to find:

∫ 8 cos y cos 2y dy

Step 1: Apply the product-to-sum formula

We know that:

cos A cos B = (1/2) [cos(A + B) + cos(A − B)]

Here, A = y and B = 2y.

So,

cos y cos 2y = (1/2) [cos(y + 2y) + cos(y − 2y)]
= (1/2) [cos 3y + cos(−y)]

Since cos(−y) = cos y, we have:

cos y cos 2y = (1/2) [cos 3y + cos y]

Step 2: Multiply by 8

8 cos y cos 2y = 8 × (1/2) [cos 3y + cos y]
= 4 [cos 3y + cos y]

Step 3: Integrate

∫ 8 cos y cos 2y dy = ∫ 4 [cos 3y + cos y] dy
= 4 ∫ cos 3y dy + 4 ∫ cos y dy
= 4 × (1/3) sin 3y + 4 sin y + C

Step 4: Final Answer

∫ 8 cos y cos 2y dy = (4/3) sin 3y + 4 sin y + C

Solved Example 3: Solve the given expression

Simplify: (cos 2x − cos 6x) / (cos 3x − cos 7x)

Solution:

We know the formula:

cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)

Step 1: Apply the formula to the numerator and denominator

Numerator: cos 2x − cos 6x

= −2 sin((2x + 6x)/2) sin((2x − 6x)/2)
= −2 sin(4x) sin(−2x)
= 2 sin 4x sin 2x (since sin(−θ) = −sin θ)

Denominator: cos 3x − cos 7x

= −2 sin((3x + 7x)/2) sin((3x − 7x)/2)
= −2 sin(5x) sin(−2x)
= 2 sin 5x sin 2x

Step 2: Simplify the fraction

(cos 2x − cos 6x) / (cos 3x − cos 7x)
= (2 sin 4x sin 2x) / (2 sin 5x sin 2x)
= sin 4x / sin 5x

Step 3: Final Answer

(cos 2x − cos 6x) / (cos 3x − cos 7x) = sin 4x / sin 5x

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