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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.

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 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 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 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

  1. 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

  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).

  3. 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

  4. 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.

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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 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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