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Product to sum identities worksheet

Two of these identities turn a product into a sum and two turn a sum into a product. They come straight out of adding and subtracting the sum and difference formulas.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Product-to-Sum Identities Date Period Rewrite or evaluate each expression using the identities. Write a product as a sum Write as a sum: $2 \cos 6 x \cos x$ Write as a sum: $2 \sin 2 x \cos 4 x$ Write as a sum: $2 \cos 4 x \cos 7 x$ Write as a sum: $2 \sin 7 x \cos x$ Write as a sum: $2 \cos x \cos 8 x$ Use sum-to-product formulas Express as a product: $\cos 18 x + \cos 8 x$ . Write as a product: $\sin 7 x - \sin 4 x$ . Express as a product: $\cos 24 x - \cos 18 x$ . Write as a product: $\cos x - \cos 5 x$ . Rewrite as a product: $\sin 6 x + \sin 2 x$ . Write a difference as a product Write as a product: $\sin 9 x - \sin 5 x$ Write as a product: $\cos 5 x - \cos 7 x$ Write as a product: $\sin x - \sin 3 x$ Write as a product: $\cos x - \cos 3 x$ Write as a product: $\sin 7 x - \sin x$ Find exact values with product-to-sum Find the exact value: $\sin 105^{\circ} - \sin 15^{\circ}$ Find the exact value: $\sin 105^{\circ} \sin 75^{\circ}$ Find the exact value: $\cos 105^{\circ} \cos 75^{\circ}$ Find the exact value: $\sin 105^{\circ} \cos 75^{\circ}$ Find the exact value: $\cos 75^{\circ} + \cos 15^{\circ}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Product-to-Sum Identities

Date Period

Rewrite or evaluate each expression using the identities.

Write a product as a sum
Write as a sum: $2 \cos 6 x \cos x$
Write as a sum: $2 \sin 2 x \cos 4 x$
Write as a sum: $2 \cos 4 x \cos 7 x$
Write as a sum: $2 \sin 7 x \cos x$
Write as a sum: $2 \cos x \cos 8 x$

Use sum-to-product formulas
Express as a product: $\cos 18 x + \cos 8 x$ .
Write as a product: $\sin 7 x - \sin 4 x$ .
Express as a product: $\cos 24 x - \cos 18 x$ .
Write as a product: $\cos x - \cos 5 x$ .
Rewrite as a product: $\sin 6 x + \sin 2 x$ .

Write a difference as a product
Write as a product: $\sin 9 x - \sin 5 x$
Write as a product: $\cos 5 x - \cos 7 x$
Write as a product: $\sin x - \sin 3 x$
Write as a product: $\cos x - \cos 3 x$
Write as a product: $\sin 7 x - \sin x$

Find exact values with product-to-sum
Find the exact value: $\sin 105^{\circ} - \sin 15^{\circ}$
Find the exact value: $\sin 105^{\circ} \sin 75^{\circ}$
Find the exact value: $\cos 105^{\circ} \cos 75^{\circ}$
Find the exact value: $\sin 105^{\circ} \cos 75^{\circ}$
Find the exact value: $\cos 75^{\circ} + \cos 15^{\circ}$



Sigma Prep · sigmaprep.io/worksheets Name Product-to-Sum IdentitiesAnswer keyVersion 1 Date Period Rewrite or evaluate each expression using the identities. Write a product as a sum Write as a sum: $2 \cos 6 x \cos x$ $\cos 5 x + \cos 7 x$ Write as a sum: $2 \sin 2 x \cos 4 x$ $\sin 6 x - \sin 2 x$ Write as a sum: $2 \cos 4 x \cos 7 x$ $\cos 3 x + \cos 11 x$ Write as a sum: $2 \sin 7 x \cos x$ $\sin 8 x + \sin 6 x$ Write as a sum: $2 \cos x \cos 8 x$ $\cos 7 x + \cos 9 x$ Use sum-to-product formulas Express as a product: $\cos 18 x + \cos 8 x$ . $2 \cos 13 x \cos 5 x$ Write as a product: $\sin 7 x - \sin 4 x$ . $2 \cos \frac{11 x}{2} \sin \frac{3 x}{2}$ Express as a product: $\cos 24 x - \cos 18 x$ . $- 2 \sin 21 x \sin 3 x$ Write as a product: $\cos x - \cos 5 x$ . $2 \sin 3 x \sin 2 x$ Rewrite as a product: $\sin 6 x + \sin 2 x$ . $2 \sin 4 x \cos 2 x$ Write a difference as a product Write as a product: $\sin 9 x - \sin 5 x$ $2 \cos 7 x \sin 2 x$ Write as a product: $\cos 5 x - \cos 7 x$ $2 \sin 6 x \sin x$ Write as a product: $\sin x - \sin 3 x$ $- 2 \cos 2 x \sin x$ Write as a product: $\cos x - \cos 3 x$ $2 \sin 2 x \sin x$ Write as a product: $\sin 7 x - \sin x$ $2 \cos 4 x \sin 3 x$ Find exact values with product-to-sum Find the exact value: $\sin 105^{\circ} - \sin 15^{\circ}$ $\frac{\sqrt{2}}{2}$ Find the exact value: $\sin 105^{\circ} \sin 75^{\circ}$ $\frac{2 + \sqrt{3}}{4}$ Find the exact value: $\cos 105^{\circ} \cos 75^{\circ}$ $\frac{\sqrt{3} - 2}{4}$ Find the exact value: $\sin 105^{\circ} \cos 75^{\circ}$ $\frac{1}{4}$ Find the exact value: $\cos 75^{\circ} + \cos 15^{\circ}$ $\frac{\sqrt{6}}{2}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Product-to-Sum IdentitiesAnswer keyVersion 1

Date Period

Rewrite or evaluate each expression using the identities.

Write a product as a sum
Write as a sum: $2 \cos 6 x \cos x$ $\cos 5 x + \cos 7 x$
Write as a sum: $2 \sin 2 x \cos 4 x$ $\sin 6 x - \sin 2 x$
Write as a sum: $2 \cos 4 x \cos 7 x$ $\cos 3 x + \cos 11 x$
Write as a sum: $2 \sin 7 x \cos x$ $\sin 8 x + \sin 6 x$
Write as a sum: $2 \cos x \cos 8 x$ $\cos 7 x + \cos 9 x$

Use sum-to-product formulas
Express as a product: $\cos 18 x + \cos 8 x$ . $2 \cos 13 x \cos 5 x$
Write as a product: $\sin 7 x - \sin 4 x$ . $2 \cos \frac{11 x}{2} \sin \frac{3 x}{2}$
Express as a product: $\cos 24 x - \cos 18 x$ . $- 2 \sin 21 x \sin 3 x$
Write as a product: $\cos x - \cos 5 x$ . $2 \sin 3 x \sin 2 x$
Rewrite as a product: $\sin 6 x + \sin 2 x$ . $2 \sin 4 x \cos 2 x$

Write a difference as a product
Write as a product: $\sin 9 x - \sin 5 x$ $2 \cos 7 x \sin 2 x$
Write as a product: $\cos 5 x - \cos 7 x$ $2 \sin 6 x \sin x$
Write as a product: $\sin x - \sin 3 x$ $- 2 \cos 2 x \sin x$
Write as a product: $\cos x - \cos 3 x$ $2 \sin 2 x \sin x$
Write as a product: $\sin 7 x - \sin x$ $2 \cos 4 x \sin 3 x$

Find exact values with product-to-sum
Find the exact value: $\sin 105^{\circ} - \sin 15^{\circ}$ $\frac{\sqrt{2}}{2}$
Find the exact value: $\sin 105^{\circ} \sin 75^{\circ}$ $\frac{2 + \sqrt{3}}{4}$
Find the exact value: $\cos 105^{\circ} \cos 75^{\circ}$ $\frac{\sqrt{3} - 2}{4}$
Find the exact value: $\sin 105^{\circ} \cos 75^{\circ}$ $\frac{1}{4}$
Find the exact value: $\cos 75^{\circ} + \cos 15^{\circ}$ $\frac{\sqrt{6}}{2}$

How to do these

$2 \sin 3 x \cos 5 x$

The identity gives sin of the sum plus sin of the difference.
The sum is 8x and the difference is minus 2x.
Sine of minus 2x is minus sin 2x.

The answer is $\sin 8 x - \sin 2 x$ .

Where students go wrong

Dropping the sign on the difference. Sine of a negative angle is negative, so sin(A − B) with B larger than A flips: cosine would not, and mixing the two rules up is the usual slip.

Also called sum to product or product to sum formulas.

What you can put on this worksheet

Product to sum: Write as a sum: $2 \sin 4 x \cos 5 x$ → $\sin 9 x - \sin x$
Sum to product: Rewrite as a product: $\sin 10 x + \sin 6 x$ . → $2 \sin 8 x \cos 2 x$
Differences to products: Write as a product: $\sin 6 x - \sin 4 x$ → $2 \cos 5 x \sin x$
Exact values: Find the exact value: $\sin 75^{\circ} \sin 45^{\circ}$ → $\frac{1 + \sqrt{3}}{4}$
A number other than 2 in front: $3 \sin 4 x \cos 3 x$ → $\frac{3}{2} (\sin 7 x + \sin x)$
Degree angles, left as trig values: $4 \sin 55^{\circ} \cos 40^{\circ}$ → $2 ⁣ (\sin 15^{\circ} + \sin 95^{\circ})$
A number outside, or degree angles: $3 (\cos 9 x - \cos 3 x)$ → $- 6 \sin 6 x \sin 3 x$
Equations solved by sum to product: Solve on $[0^{\circ}, 360^{\circ})$ : $\cos 4 x + \cos 2 x = 0$ → $30^{\circ},$ $90^{\circ},$ $150^{\circ},$ $210^{\circ},$ $270^{\circ},$ $330^{\circ}$

Questions about these worksheets

Yes. Every product to sum identities sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the product to sum identities questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 8: product to sum, sum to product, differences to products, exact values, a number other than 2 in front, degree angles, left as trig values, a number outside, or degree angles and equations solved by sum to product. Tick as many as you want and set how many of each, or let it spread them evenly.

Dropping the sign on the difference. Sine of a negative angle is negative, so sin(A − B) with B larger than A flips: cosine would not, and mixing the two rules up is the usual slip.

Product to sum identities worksheet

Select your difficulty

Write a product as a sum

Use sum-to-product formulas

Write a difference as a product

Find exact values with product-to-sum

Write a product as a sum

Use sum-to-product formulas

Write a difference as a product

Find exact values with product-to-sum

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these product to sum identities worksheets free?+

Do the product to sum identities worksheets come with an answer key?+

Are these product to sum identities worksheets printable?+

What grade are these product to sum identities worksheets for?+

Can I get a different version of the same product to sum identities worksheet?+

Is this a free alternative to Kuta Software for product to sum identities?+

Can I choose how hard the product to sum identities questions are?+

What kinds of question are on a product to sum identities worksheet?+

What do students get wrong with product to sum identities?+