Sum and Difference Formulas - Proof, Application | Sum and Difference Identities (2024)

The sum and difference formulas in trigonometry are used to find the value of the trigonometric functions at specific angles where it is easier to express the angle as the sum or difference of unique angles (0°, 30°, 45°, 60°, 90°, and 180°). We memorize the values of trigonometric functions at 0°, 30°, 45°, 60°, 90°, and 180°. So, to find the value of a trigonometric function at 105°, we can write it as 105° = 45° + 60° and simplify the problem. Mainly, we have the sum and difference identities of sine, cosine, and tangent functions. The sum and difference formulas have many applications including finding the distance of the Earth from the Sun and measuring the height of a mountain.

The sum and difference identities are used to solve various mathematical problems and prove the trigonometric formulas and identities. In this article, we will discuss the sum and difference formulas for sine, cosine, and tangent functions and prove the identities using trigonometric formulas. Also, we will learn to use these formulas with the help of solved examples for a better understanding and summarize these identities in a table for a quick review and revision. Please note that we will be using the terms formulas and identities interchangeably throughout the article.

What are Sum and Difference Formulas?

We have six main sum and difference formulas for the trigonometric functions including the sine function, cosine function, and tangent function. These formulas help us to evaluate the value of the trigonometric functions at angles which can be expressed as the sum or difference of special angles 0°, 30°, 45°, 60°, 90°, and 180°. The list of sum and difference formulas is as follows:

• sin(A + B) = sinA cosB + cosA sinB
• sin(A - B) = sinA cosB - cosA sinB
• cos(A + B) = cosA cosB - sinA sinB
• cos(A - B) = cosA cosB + sinA sinB
• tan(A + B) = (tanA + tanB) / (1 - tanA tanB)
• tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

Let us now prove the above-mentioned sum and difference identities in the next section.

Proof of Sum and Difference Identities

As we discussed in the previous section, we have mainly six sum and difference identities of the sine function, cosine function, and tangent function. Now, we will prove them one by one. Consider a unit circle in a coordinate plane. We know that the coordinates of points on the unit circle are given by, (cosθ, sinθ). Now, in the unit circle given below, point P makes an angle α with the positive x-axis and has coordinates (cos α, sin α) whereas the point Q makes an angle β with the positive x-axis and has coordinates (cos β, sin β). Observe that the angle POQ is equal to α - β.

Next, label two points A and B on the circle such that the point B is on the x-axis with coordinates (1, 0) and angle AOB is equal to α - β and hence, point A has coordinates (cos (α - β), sin (α - β)). Note that triangle POQ and AOB are rotations of one another and hence, the lengths of PQ and AB are the same, that is, the distance from P to Q is equal to the distance from A to B.

Sum and Difference Formulas for Cosine

First, we will prove the difference formula for the cosine function. Since PQ is equal to AB, so using the distance formula, the distance between the points P and Q is given by,

d_PQ = √[(cos α - cos β)² + (sin α - sin β)²]

= √[cos²α - 2 cos α cos β + cos²β + sin²α - 2 sin α sin β + sin²β] --- [Using algebraic identity (a - b)2 = a2 - 2ab + b2]

= √[(cos²α + sin²α) + (cos²β + sin²β) - 2 cos α cos β - 2 sin α sin β] --- [Using trigonometric formula cos²A + sin²A = 1]

= √[1 + 1 - 2 cos α cos β - 2 sin α sin β]

= √(2 - 2 cos α cos β - 2 sin α sin β) --- (1)

Now, the distance between the points A and B is given by,

d_AB = √[(cos (α - β) - 1)² + (sin (α - β) - 0)²]

= √[cos²(α - β) + 1 - 2 cos (α - β) + sin²(α - β)]

= √(cos²(α - β) + sin²(α - β) + 1 - 2 cos (α - β))

= √(1 + 1 - 2 cos (α - β))

= √[2 - 2 cos (α - β)] --- (2)

Since the distance between P and Q is equal to the distance between A and B, we have d_PQ = d_AB. So, from (1) and (2), we have

√(2 - 2 cos α cos β - 2 sin α sin β) = √[2 - 2 cos (α - β)]

Squaring both sides of the above equation,

⇒ 2 - 2 cos α cos β - 2 sin α sin β = 2 - 2 cos (α - β)

⇒ 2 [1 - cos α cos β - sin α sin β] = 2 [1 - cos (α - β)]

⇒ 1 - cos α cos β - sin α sin β = 1 - cos (α - β)

⇒ - cos α cos β - sin α sin β = - cos (α - β)

⇒ cos (α - β) = cos α cos β + sin α sin β

Hence, we have proved the difference formula of the cosine function.

We can derive the sum formula for the cosine function by substituting β = - (-β) into the difference formula of cos. So, we have

cos (α + β) = cos [α - (-β)]

= cos α cos (-β) + sin α sin (-β) --- [Using the formula cos (α - β) = cos α cos β + sin α sin β]

= cos α cos β + sin α (-sin β) --- [Because cos(-x) = cos x and sin(-x) = -sin x]

= cos α cos β - sin α sin β

⇒ cos (α + β) = cos α cos β - sin α sin β

Thus, we have derived the sum and difference formulas for the cosine function.

Sum and Difference Formulas for Sine

Next, we will prove the sum and difference for the sine function using the cofunction identities of trigonometry. We know that sin A = cos (π/2 - A) and cos A = sin (π/2 - A). So, we have

sin (α + β) = cos [π/2 - (α + β)]

= cos [π/2 - α - β]

= cos [(π/2 - α) - β]

= cos (π/2 - α) cos β + sin (π/2 - α) sin β

= sin α cos β + cos α sin β

⇒ sin (α + β) = sin α cos β + cos α sin β

Now, we can write α - β = α + (-β). So, using the sum formula for the sine function, we have

sin (α - β) = sin [α + (-β)]

= sin α cos (-β) + cos α sin (-β)

= sin α cos β + cos α (-sin β) --- [Because cos (-x) = cos x and sin (-x) = - sin x]

= sin α cos β - cos α sin β

⇒ sin (α - β) = sin α cos β - cos α sin β

Hence, we have proved the sum and difference formulas for the sine function.

Sum and Difference Formulas for Tangent

In this section, we will prove the sum and difference identities for the tangent function. We know that tangent function can be written as the ratio of the sine and cosine, that is, tan A = sin A / cos A. So, we can write tan (α + β) as,

tan (α + β) = sin (α + β) / cos (α + β)

= (sin α cos β + cos α sin β) / (cos α cos β - sin α sin β) --- [Using sin (α + β) = sin α cos β + cos α sin β and cos (α + β) = cos α cos β - sin α sin β]

Dividing the numerator and denominator by cos α cos β, we have

= [ (sin α cos β + cos α sin β) / cos α cos β] / [ (cos α cos β - sin α sin β) / cos α cos β ]

= (sin α / cos α + sin β / cos β ) / [1 - (sin α / cos α) × (sin β / cos β)]

= (tan α + tan β) / (1 - tan α tan β)

⇒ tan (α + β) = (tan α + tan β) / (1 - tan α tan β)

Now, using the above formula, we can find the formula for tan (α - β). So, we have

tan (α - β) = tan [α + (-β)]

= [tan α + tan (-β)] / [1 - tan α tan (-β)] --- [Using tan (A + B) formula]

= [tan α + (-tan β)] / [1 - tan α (-tan β)] --- [Because tan (-x) = - tan x]

= (tan α - tan β) / (1 + tan α tan β)

⇒ tan (α - β) = (tan α - tan β) / (1 + tan α tan β)

Hence, we have proved the sum and difference formulas of the tangent function.

Sum and Difference Identities Table

In the previous section, we derived the formulas of all the sum and difference identities of the trigonometric functions sine, cosine, and tangent. Now, let us summarize these formulas in the table below for a quick revision.

How to Apply Sum and Difference Formulas

Now that we have understood the sum and difference formulas of trigonometric functions, let us now use these formulas to understand their application. We can simply substitute the values of the angles into the formulas or express the given angle as a sum or difference of special angles to find their values. The sum and difference identities also help us to verify various trigonometric formulas and identities. Let us solve a few examples given below and learn to apply these identities:

Example 1: Evaluate the value of sin (5π/4 - π/6)

Solution: Using the difference formula of sine, we have

sin (5π/4 - π/6) = sin5π/4 cos π/6 - cos5π/4 sin π/6 --- [Using sin(A - B) = sinA cosB - cosA sinB]

= (-1/√2) (√3/2) - (-1/√2) (1/2)

= -√3/2√2 + 1/2√2

= (1 - √3) / 2√2

Example 2: Find the value of cos 105°.

Solution: We can write 105° as 105° = 60° + 45°. So, using the sum formula of cosine, we have

cos 105° = cos (60° + 45°)

= cos 60° cos 45° - sin 60° sin 45° --- [Using cos(A + B) = cosA cosB - sinA sinB]

= (1/2) (1/√2) - (√3/2) (1/√2)

= 1/2√2 - √3/2√2

= (1 - √3) / 2√2

Example 3: Find the value of tan (2π/3 + π/4).

Solution: To find the value of tan (2π/3 + π/4), we will use the sum formula of the tangent function.

tan (2π/3 + π/4) = (tan2π/3 + tanπ/4) / (1 - tan2π/3 tanπ/4) --- [Using tan(A + B) = (tanA + tanB) / (1 - tanA tanB)]

= [(-√3) + 1] / [1 - (-√3) (1)]

= (1 - √3) / (1 + √3)

Important Notes on Sum and Difference Formulas

• The six important sum and difference formulas are:
  • sin(A + B) = sinA cosB + cosA sinB
  • sin(A - B) = sinA cosB - cosA sinB
  • cos(A + B) = cosA cosB - sinA sinB
  • cos(A - B) = cosA cosB + sinA sinB
  • tan(A + B) = (tanA + tanB) / (1 - tanA tanB)
  • tan(A - B) = (tanA - tanB) / (1 + tanA tanB)
• We can derive these formulas using a unit circle and trigonometric formulas.
• The sum and difference identities are used to find the value of trigonometric functions at angles that can be written as the sum or difference of the special angles 0°, 30°, 45°, 60°, 90°, and 180°.

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