Sum-to-Product and Product-to-Sum Formulas

Photo of the UCLA marching band. — Figure 1 The UCLA marching band (credit: Eric Chan, Flickr).

A band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that can be interpreted using trigonometric functions. For example, Figure 2 represents a sound wave for the musical note A. In this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves.

Graph of a sound wave for the musical note A - it is a periodic function much like sin and cos - from 0 to .01 — Figure 2

Expressing Products as Sums

We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas, which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.

Expressing Products as Sums for Cosine

We can derive the product-to-sum formula from the sum and difference identities for cosine. If we add the two equations, we get:

\begin{array}{l} \underset{________________________________}{\begin{matrix} \cos α \cos β + \sin α \sin β = \cos (α - β) \\ + \cos α \cos β - \sin α \sin β = \cos (α + β) \end{matrix}} \\ 2 \cos α \cos β = \cos (α - β) + \cos (α + β) \end{array}

Then, we divide by $2$ to isolate the product of cosines:

\cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)]

Example 1

Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine

Write the following product of cosines as a sum: $2 \cos (\frac{7 x}{2}) \cos \frac{3 x}{2} .$

Solution

We begin by writing the formula for the product of cosines:

\cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)]

We can then substitute the given angles into the formula and simplify.

\begin{array}{l} 2 \cos (\frac{7 x}{2}) \cos (\frac{3 x}{2}) = (2) (\frac{1}{2}) [\cos (\frac{7 x}{2} - \frac{3 x}{2}) + \cos (\frac{7 x}{2} + \frac{3 x}{2})] \\ = [\cos (\frac{4 x}{2}) + \cos (\frac{10 x}{2})] \\ = \cos 2 x + \cos 5 x \end{array}

Expressing the Product of Sine and Cosine as a Sum

Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine. If we add the sum and difference identities, we get:

\begin{array}{l} \underset{_________________________________________}{\begin{array}{l} \begin{array}{l} \sin (α + β) = \sin α \cos β + \cos α \sin β \end{array} \\ + \sin (α - β) = \sin α \cos β - \cos α \sin β \end{array}} \\ \sin (α + β) + \sin (α - β) = 2 \sin α \cos β \end{array}

Then, we divide by 2 to isolate the product of cosine and sine:

\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)]

Example 2

Writing the Product as a Sum Containing only Sine or Cosine

Express the following product as a sum containing only sine or cosine and no products: $\sin (4 θ) \cos (2 θ) .$

Solution

Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.

\begin{array}{l} \sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin (4 θ) \cos (2 θ) = \frac{1}{2} [\sin (4 θ + 2 θ) + \sin (4 θ - 2 θ)] \\ = \frac{1}{2} [\sin (6 θ) + \sin (2 θ)] \end{array}

Expressing Products of Sines in Terms of Cosine

Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:

\begin{array}{l} \underset{____________________________________________________}{\begin{array}{l} \begin{array}{l} \cos (α - β) = \cos α \cos β + \sin α \sin β \end{array} \\ - \cos (α + β) = - (\cos α \cos β - \sin α \sin β) \end{array}} \\ \cos (α - β) - \cos (α + β) = 2 \sin α \sin β \end{array}

Then, we divide by 2 to isolate the product of sines:

\sin α \sin β = \frac{1}{2} [\cos (α - β) - \cos (α + β)]

Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.

The Product-to-Sum Formulas

The product-to-sum formulas are as follows:

\cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)]

\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)]

\sin α \sin β = \frac{1}{2} [\cos (α - β) - \cos (α + β)]

\cos α \sin β = \frac{1}{2} [\sin (α + β) - \sin (α - β)]

Example 3

Express the Product as a Sum or Difference

Write $\cos (3 θ) \cos (5 θ)$ as a sum or difference.

Solution

We have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.

\begin{array}{l} \cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \cos (3 θ) \cos (5 θ) = \frac{1}{2} [\cos (3 θ - 5 θ) + \cos (3 θ + 5 θ)] \\ = \frac{1}{2} [\cos (2 θ) + \cos (8 θ)] & Use even-odd identity . \end{array}

Expressing Sums as Products

Some problems require the reverse of the process we just used. The sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for sine. Let $\frac{u + v}{2} = α$ and $\frac{u - v}{2} = β .$

Then,

\begin{array}{l} α + β = \frac{u + v}{2} + \frac{u - v}{2} \\ = \frac{2 u}{2} \\ = u \\ α - β = \frac{u + v}{2} - \frac{u - v}{2} \\ = \frac{2 v}{2} \\ = v \end{array}

Thus, replacing $α$ and $β$ in the product-to-sum formula with the substitute expressions, we have

\begin{array}{l} \sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin (\frac{u + v}{2}) \cos (\frac{u - v}{2}) = \frac{1}{2} [\sin u + \sin v] & Substitute for (α + β) and (α - β) \\ 2 \sin (\frac{u + v}{2}) \cos (\frac{u - v}{2}) = \sin u + \sin v \end{array}

The other sum-to-product identities are derived similarly.

Sum-to-Product Formulas

The sum-to-product formulas are as follows:

\sin α + \sin β = 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2})

\sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2})

\cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2})

\cos α + \cos β = 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2})

Example 4

Writing the Difference of Sines as a Product

Write the following difference of sines expression as a product: $\sin (4 θ) - \sin (2 θ) .$

Solution

We begin by writing the formula for the difference of sines.

\sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2})

Substitute the values into the formula, and simplify.

\begin{array}{l} \sin (4 θ) - \sin (2 θ) = 2 \sin (\frac{4 θ - 2 θ}{2}) \cos (\frac{4 θ + 2 θ}{2}) \\ = 2 \sin (\frac{2 θ}{2}) \cos (\frac{6 θ}{2}) \\ = 2 \sin θ \cos (3 θ) \end{array}

Example 5

Evaluating Using the Sum-to-Product Formula

Evaluate $\cos (15^{\circ}) - \cos (75^{\circ}) .$

Solution

We begin by writing the formula for the difference of cosines.

\cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2})

Then we substitute the given angles and simplify.

\begin{array}{l} \cos (15^{\circ}) - \cos (75^{\circ}) = - 2 \sin (\frac{15^{\circ} + 75^{\circ}}{2}) \sin (\frac{15^{\circ} - 75^{\circ}}{2}) \\ = - 2 \sin (45^{\circ}) \sin (- 30^{\circ}) \\ = - 2 (\frac{\sqrt{2}}{2}) (- \frac{1}{2}) \\ = \frac{\sqrt{2}}{2} \end{array}

Example 6

Proving an Identity

Prove the identity:

\frac{\cos (4 t) - \cos (2 t)}{\sin (4 t) + \sin (2 t)} = - \tan t

Solution

We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.

\begin{array}{l} \frac{\cos (4 t) - \cos (2 t)}{\sin (4 t) + \sin (2 t)} = \frac{- 2 \sin (\frac{4 t + 2 t}{2}) \sin (\frac{4 t - 2 t}{2})}{2 \sin (\frac{4 t + 2 t}{2}) \cos (\frac{4 t - 2 t}{2})} \\ = \frac{- 2 \sin (3 t) \sin t}{2 \sin (3 t) \cos t} \\ = \frac{- 2 \sin (3 t) \sin t}{2 \sin (3 t) \cos t} \\ = - \frac{\sin t}{\cos t} \\ = - \tan t \end{array}

Analysis

Recall that verifying trigonometric identities has its own set of rules. The procedures for solving an equation are not the same as the procedures for verifying an identity. When we prove an identity, we pick one side to work on and make substitutions until that side is transformed into the other side.

Example 7

Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities

Verify the identity $\csc^{2} θ - 2 = \frac{\cos (2 θ)}{\sin^{2} θ} .$

Solution

For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.

\begin{array}{l} \frac{\cos (2 θ)}{\sin^{2} θ} = \frac{1 - 2 \sin^{2} θ}{\sin^{2} θ} \\ = \frac{1}{\sin^{2} θ} - \frac{2 \sin^{2} θ}{\sin^{2} θ} \\ = \csc^{2} θ - 2 \end{array}

Key Equations

..
Product-to-sum Formulas	$\begin{array}{l} \cos α \cos β = \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin α \sin β = \frac{1}{2} [\cos (α - β) - \cos (α + β)] \\ \cos α \sin β = \frac{1}{2} [\sin (α + β) - \sin (α - β)] \end{array}$
Sum-to-product Formulas	$\begin{array}{l} \sin α + \sin β = 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \\ \sin α - \sin β = 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2}) \\ \cos α - \cos β = - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2}) \\ \cos α + \cos β = 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \end{array}$

Key Concepts

From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See Example 1, Example 2, and Example 3.
We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See Example 4.
Trigonometric expressions are often simpler to evaluate using the formulas. See Example 5.
The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See Example 6 and Example 7.

Section Exercises

Verbal

Exercise 1

Starting with the product to sum formula $\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)],$ explain how to determine the formula for $\cos α \sin β .$

Solution

Substitute $α$ into cosine and $β$ into sine and evaluate.

Exercise 2

Explain two different methods of calculating $\cos (195°) \cos (105°),$ one of which uses the product to sum. Which method is easier?

Exercise 3

Explain a situation where we would convert an equation from a sum to a product and give an example.

Solution

Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: $\frac{\sin (3 x) + \sin x}{\cos x} = 1.$ When converting the numerator to a product the equation becomes: $\frac{2 \sin (2 x) \cos x}{\cos x} = 1$

Exercise 4

Explain a situation where we would convert an equation from a product to a sum, and give an example.

Algebraic

For the following exercises, rewrite the product as a sum or difference.

Exercise 5

$16 \sin (16 x) \sin (11 x)$

Solution

$8 (\cos (5 x) - \cos (27 x))$

Exercise 6

$20 \cos (36 t) \cos (6 t)$

Exercise 7

$2 \sin (5 x) \cos (3 x)$

Solution

$\sin (2 x) + \sin (8 x)$

Exercise 8

$10 \cos (5 x) \sin (10 x)$

Exercise 9

$\sin (- x) \sin (5 x)$

Solution

$\frac{1}{2} (\cos (6 x) - \cos (4 x))$

Exercise 10

$\sin (3 x) \cos (5 x)$

For the following exercises, rewrite the sum or difference as a product.

Exercise 11

$\cos (6 t) + \cos (4 t)$

Solution

$2 \cos (5 t) \cos t$

Exercise 12

$\sin (3 x) + \sin (7 x)$

Exercise 13

$\cos (7 x) + \cos (- 7 x)$

Solution

$2 \cos (7 x)$

Exercise 14

$\sin (3 x) - \sin (- 3 x)$

Exercise 15

$\cos (3 x) + \cos (9 x)$

Solution

$2 \cos (6 x) \cos (3 x)$

Exercise 16

$\sin h - \sin (3 h)$

For the following exercises, evaluate the product for the following using a sum or difference of two functions.

Exercise 17

$\cos (45°) \cos (15°)$

Solution

$\frac{1}{4} (1 + \sqrt{3})$

Exercise 18

$\cos (45°) \sin (15°)$

Exercise 19

$\sin (−345°) \sin (−15°)$

Solution

$\frac{1}{4} (\sqrt{3} - 2)$

Exercise 20

$\sin (195°) \cos (15°)$

Exercise 21

$\sin (−45°) \sin (−15°)$

Solution

$\frac{1}{4} (\sqrt{3} - 1)$

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

Exercise 22

$\cos (23°) \sin (17°)$

Exercise 23

$2 \sin (100°) \sin (20°)$

Solution

$\cos (80°) - \cos (120°)$

Exercise 24

$2 \sin (−100°) \sin (−20°)$

Exercise 25

$\sin (213°) \cos (8°)$

Solution

$\frac{1}{2} (\sin (221°) + \sin (205°))$

Exercise 26

$2 \cos (56°) \cos (47°)$

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

Exercise 27

$\sin (76°) + \sin (14°)$

Solution

$\sqrt{2} \cos (31°)$

Exercise 28

$\cos (58°) - \cos (12°)$

Exercise 29

$\sin (101°) - \sin (32°)$

Solution

$2 \cos (66.5 °) \sin (34.5 °)$

Exercise 30

$\cos (100°) + \cos (200°)$

Exercise 31

$\sin (−1°) + \sin (−2°)$

Solution

$2 \sin (−1.5°) \cos (0.5°)$

For the following exercises, prove the identity.

Exercise 32

$\frac{\cos (a + b)}{\cos (a - b)} = \frac{1 - \tan a \tan b}{1 + \tan a \tan b}$

Exercise 33

$4 \sin (3 x) \cos (4 x) = 2 \sin (7 x) - 2 \sin x$

Solution

$\begin{array}{l} 2 \sin (7 x) - 2 \sin x = 2 \sin (4 x + 3 x) - 2 \sin (4 x - 3 x) = \\ 2 (\sin (4 x) \cos (3 x) + \sin (3 x) \cos (4 x)) - 2 (\sin (4 x) \cos (3 x) - \sin (3 x) \cos (4 x)) = \\ 2 \sin (4 x) \cos (3 x) + 2 \sin (3 x) \cos (4 x)) - 2 \sin (4 x) \cos (3 x) + 2 \sin (3 x) \cos (4 x)) = \\ 4 \sin (3 x) \cos (4 x) \end{array}$

Exercise 34

$\frac{6 \cos (8 x) \sin (2 x)}{\sin (- 6 x)} = −3 \sin (10 x) \csc (6 x) + 3$

Exercise 35

$\sin x + \sin (3 x) = 4 \sin x \cos^{2} x$

Solution

$\sin x + \sin (3 x) = 2 \sin (\frac{4 x}{2}) \cos (\frac{- 2 x}{2}) =$
$2 \sin (2 x) \cos x = 2 (2 \sin x \cos x) \cos x =$
$4 \sin x \cos^{2} x$

Exercise 36

$2 (\cos^{3} x - \cos x \sin^{2} x) = \cos (3 x) + \cos x$

Exercise 37

$2 \tan x \cos (3 x) = \sec x (\sin (4 x) - \sin (2 x))$

Solution

$2 \tan x \cos (3 x) = \frac{2 \sin x \cos (3 x)}{\cos x} = \frac{2 (.5 (\sin (4 x) - \sin (2 x)))}{\cos x}$
$= \frac{1}{\cos x} (\sin (4 x) - \sin (2 x)) = \sec x (\sin (4 x) - \sin (2 x))$

Exercise 38

$\cos (a + b) + \cos (a - b) = 2 \cos a \cos b$

Numeric

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

Exercise 39

$\cos (58^{\circ}) + \cos (12^{\circ})$

Solution

$2 \cos (35^{\circ}) \cos (23^{\circ}), 1 .5081$

Exercise 40

$\sin (2^{\circ}) - \sin (3^{\circ})$

Exercise 41

$\cos (44^{\circ}) - \cos (22^{\circ})$

Solution

$- 2 \sin (33^{\circ}) \sin (11^{\circ}), - 0.2078$

Exercise 42

$\cos (176^{\circ}) \sin (9^{\circ})$

Exercise 43

$\sin (- 14^{\circ}) \sin (85^{\circ})$

Solution

$\frac{1}{2} (\cos (99^{\circ}) - \cos (71^{\circ})), - 0.2410$

Technology

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

Exercise 44

$2 \sin (2 x) \sin (3 x) = \cos x - \cos (5 x)$

Exercise 45

$\frac{\cos (10 θ) + \cos (6 θ)}{\cos (6 θ) - \cos (10 θ)} = \cot (2 θ) \cot (8 θ)$

Solution

It is and identity.

Exercise 46

$\frac{\sin (3 x) - \sin (5 x)}{\cos (3 x) + \cos (5 x)} = \tan x$

Exercise 47

$2 \cos (2 x) \cos x + \sin (2 x) \sin x = 2 \sin x$

Solution

It is not an identity, but $2 \cos^{3} x$ is.

Exercise 48

$\frac{\sin (2 x) + \sin (4 x)}{\sin (2 x) - \sin (4 x)} = - \tan (3 x) \cot x$

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

Exercise 49

$\frac{\sin (9 t) - \sin (3 t)}{\cos (9 t) + \cos (3 t)}$

Solution

$\tan (3 t)$

Exercise 50

$2 \sin (8 x) \cos (6 x) - \sin (2 x)$

Exercise 51

$\frac{\sin (3 x) - \sin x}{\sin x}$

Solution

$2 \cos (2 x)$

Exercise 52

$\frac{\cos (5 x) + \cos (3 x)}{\sin (5 x) + \sin (3 x)}$

Exercise 53

$\sin x \cos (15 x) - \cos x \sin (15 x)$

Solution

$- \sin (14 x)$

Extensions

For the following exercises, prove the following sum-to-product formulas.

Exercise 54

$\sin x - \sin y = 2 \sin (\frac{x - y}{2}) \cos (\frac{x + y}{2})$

Exercise 55

$\cos x + \cos y = 2 \cos (\frac{x + y}{2}) \cos (\frac{x - y}{2})$

Solution

Start with $\cos x + \cos y .$ Make a substitution and let $x = α + β$ and let $y = α - β,$ so $\cos x + \cos y$ becomes
$\cos (α + β) + \cos (α - β) = \cos α \cos β - \sin α \sin β + \cos α \cos β + \sin α \sin β = 2 \cos α \cos β$

Since $x = α + β$ and $y = α - β,$ we can solve for $α$ and $β$ in terms of x and y and substitute in for $2 \cos α \cos β$ and get $2 \cos (\frac{x + y}{2}) \cos (\frac{x - y}{2}) .$

For the following exercises, prove the identity.

Exercise 56

$\frac{\sin (6 x) + \sin (4 x)}{\sin (6 x) - \sin (4 x)} = \tan (5 x) \cot x$

Exercise 57

$\frac{\cos (3 x) + \cos x}{\cos (3 x) - \cos x} = - \cot (2 x) \cot x$

Solution

$\frac{\cos (3 x) + \cos x}{\cos (3 x) - \cos x} = \frac{2 \cos (2 x) \cos x}{- 2 \sin (2 x) \sin x} = - \cot (2 x) \cot x$

Exercise 58

$\frac{\cos (6 y) + \cos (8 y)}{\sin (6 y) - \sin (4 y)} = \cot y \cos (7 y) \sec (5 y)$

Exercise 59

$\frac{\cos (2 y) - \cos (4 y)}{\sin (2 y) + \sin (4 y)} = \tan y$

Solution

$\begin{array}{l} \frac{\cos (2 y) - \cos (4 y)}{\sin (2 y) + \sin (4 y)} = \frac{- 2 \sin (3 y) \sin (- y)}{2 \sin (3 y) \cos y} = \\ \frac{2 \sin (3 y) \sin (y)}{2 \sin (3 y) \cos y} = \tan y \end{array}$

Exercise 60

$\frac{\sin (10 x) - \sin (2 x)}{\cos (10 x) + \cos (2 x)} = \tan (4 x)$

Exercise 61

$\cos x - \cos (3 x) = 4 \sin^{2} x \cos x$

Solution

$\begin{array}{l} \cos x - \cos (3 x) = - 2 \sin (2 x) \sin (- x) = \\ 2 (2 \sin x \cos x) \sin x = 4 \sin^{2} x \cos x \end{array}$

Exercise 62

${(\cos (2 x) - \cos (4 x))}^{2} + {(\sin (4 x) + \sin (2 x))}^{2} = 4 \sin^{2} (3 x)$

Exercise 63

$\tan (\frac{π}{4} - t) = \frac{1 - \tan t}{1 + \tan t}$

Solution

$\tan (\frac{π}{4} - t) = \frac{\tan (\frac{π}{4}) - \tan t}{1 + \tan (\frac{π}{4}) \tan (t)} = \frac{1 - \tan t}{1 + \tan t}$

product-to-sum formula

a trigonometric identity that allows the writing of a product of trigonometric functions as a sum or difference of trigonometric functions

sum-to-product formula

a trigonometric identity that allows, by using substitution, the writing of a sum of trigonometric functions as a product of trigonometric functions