Sum to Product Formulas

The sum to product formulas are trigonometric identities that convert the sum or difference of two trigonometric functions into a product of trigonometric functions. These formulas are particularly useful in simplifying expressions, solving trigonometric equations, and integrating functions.

Sum to Product formulas are important formulas of trigonometry. Four sum-to-product formulas in trigonometry are,

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

In this article, we will learn about Sum to Product Formulas, Proof of Sum to Product Formulas, Application of Sum to Product Formulas in detail.

Trigonometry Identities

Identities that give the relation between different trigonometric ratios are called Trigonometric Identities. These identities relate all trigonometric ratios with each other. There are multiple trigonometric identities including sum to product formulas in trigonometry.

Sum to Product Formulas

Sum-to-product formulas in trigonometry convert the sum of sine and cosine functions to product form. They help to easily solve the sum problems the sum-to-product formulas are,

Formula of sin A plus sin B, i.e. (sin A + sin B)

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

Formula of sin A minus sin B, i.e. (sin A - sin B)

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

Formula of cos A plus cos B, i.e. (cos A + cos B)

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

Formula of cos A minus cos B, i.e. (cos A - cos B)

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

Sum to Product Formula List

Below is the list of Sum to Product Formula List.

Sum to Product Formulas for Sine

Sum to product formulas for sine are listed below:

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

Sum to Product Formulas for Cosine

Sum to product formulas for cosine are listed below:

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

Proof of Sum to Product Formulas

Proof of Sum to Product formulas are added below,

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

From the product to sum formula we can write:

2 sin P cos Q = sin (P + Q) + sin (P - Q)

Putting P = [(A+B)/2] and Q = [(A-B)/2]

Substituting these values in above equation we get,

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

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

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

Hence Proved

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

From the product to sum formula we can write:

2 cos P sin Q = sin (P + Q) - sin (P - Q)

Putting P = [(A+B)/2] and Q = [(A-B)/2]

Substituting these values in above equation we get,

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

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

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

Hence Proved

Proof of cos A + cos B = 2 cos[(A+B)/2] cos[(A-B)/2]

From the product to sum formula we can write:

2 cos P cos Q = cos (P + Q) + cos (P - Q)

Putting P = [(A+B)/2] and Q = [(A-B)/2]

Substituting these values in above equation we get,

2 cos[(A+B)/2] cos [(A-B)/2] = cos[{(A+B)/2} + {(A-B)/2}] + cos[{(A+B)/2} - {(A-B)/2}]

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

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

Hence Proved

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

From the product to sum formula we can write:

2 sin P sin Q = cos (P + Q) - cos (P - Q)

Putting P = [(A+B)/2] and Q = [(A-B)/2]

Substituting these values in above equation we get,

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

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

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

Hence Proved

Sum to Product Formulas for Hyperbolic Functions

Sum to product formulas for hyperbolic functions are listed below:

• sinh A + sinh B = 2 sinh[(A+B)/2] cosh[(A-B)/2]
• sinh A - sinh B = 2 cosh[(A+B)/2] sinh[(A-B)/2]
• cosh A + cosh B = 2 cosh[(A+B)/2] cosh[(A-B)/2]
• cosh A - cosh B = 2 sinh[(A+B)/2] sinh[(A-B)/2]

Summary on Sum to Product Formula

• Sum to product formulas is used to find expression for sum and difference of sines and cosines functions as products of sine and cosine functions.
• Sum to product formulas in trigonometry are:
  • sin A + sin B = 2 sin [(A + B)/2] cos [(A - B)/2]
  • sin A - sin B = 2 sin [(A - B)/2] cos [(A + B)/2]
  • cos A - cos B = -2 sin [(A + B)/2] sin [(A - B)/2]
  • cos A + cos B = 2 cos [(A + B)/2] cos [(A - B)/2]
• To derive sum to product formula we use product to sum formulas in trigonometry.
• These formulas helps to simplify trigonometric problems.

Examples of Sum to Product Formulas

Example 1: Evaluate cos 155° - cos 25°.

Solution:

We know that,

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

cos 155° - cos 25° = 2 sin [(155°+25°)/2] sin [(155°-25°)/2]

⇒ cos 155° - cos 25° = 2 sin [180°/2] sin [130°/2]

⇒ cos 155° - cos 25° = 2 × sin 90° × sin 65°

⇒ cos 155° - cos 25° = 2 × 1 × 0.9

Thus, cos 155° - cos 25° = 1.8

Example 2: Solve [sin 4a + sin 2a] /cos a.

Solution:

We know that,

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

sin 4a + sin 2a = 2 sin[(4a+2a)/2] cos[(4a-2a)/2]

⇒ sin 4a + sin 2a = 2 sin[6a/2] cos[2a/2]

⇒ sin 4a + sin 2a = 2 sin 3a cos a

⇒ [sin 4a + sin 2a] /cos a = [2 sin 3a cos a] /cos a

Thus, [sin 4a + sin 2a] /cos a = 2 sin 3a

Example 3: Prove that [sin 10x - sin 4x] / [cos 12x - cos 6x] = cos 7x / sin 9x

Solution:

We know that,

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

sin 10x - sin 4x = 2 cos [(10x + 4x)/2] sin [(10x - 4x)/2]

sin 10x - sin 4x = 2 cos [14x/2] sin [6x/2]

⇒ sin 10x - sin 4x = 2 cos 7x sin 3x

⇒ cos 12x - cos 6x = 2 sin [(12x + 6x)/2] sin [(12x - 6x)/2]

⇒ cos 12x - cos 6x = 2 sin [18x/2] sin [6x/2]

⇒ cos 12x - cos 6x = 2 sin 9x sin 3x

⇒ [sin 10x - sin 4x] / [cos 12x - cos 6x] = [2 cos 7x sin 3x] / [2 sin 9x sin 3x]

Thus, [sin 10x - sin 4x] / [cos 12x - cos 6x] = cos 7x / sin 9x

Example 4: Find cos 15x + cos 3x

Solution:

We know that,

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

cos 15x + cos 3x = 2 cos [(15x + 3x)/2] cos [(15x - 3x)/2]

⇒ cos 15x + cos 3x = 2 cos [18x/2] cos [12x/2]

Thus, cos 15x + cos 3x = 2 cos 9x cos 6x

Practice Problems on Sum to Product Formulas

Problem 1: Solve: sin 10y - sin 6y.

Problem 2: Prove that: (cos 8x + cos 6x) / sin 2x = 2 cot 2x cos 14x.

Problem 3: Evaluate: (cos 4a - cos 2a) / (cos 4a + cos 2a).

Problem 4: Evaluate: sin 15° + sin 45°.