Product-to-Sum and Sum-to-Product Identities
Advanced techniques for simplifying trigonometric expressions
Product-to-sum and sum-to-product identities are powerful tools for rewriting trigonometric expressions. They can turn a product such as sin(A)cos(B) into a sum, or combine a sum such as sin(A) + sin(B) into a product. This often reveals special angles and makes exact answers much easier to find.
The Main Idea
Use product-to-sum identities when two trigonometric functions are multiplied. Use sum-to-product identities when two sine or cosine terms are added or subtracted.
Product-to-Sum Identities
Sine-Cosine Products
Cosine Products
Example 1: Convert a Product to a Sum
Simplify sin(75°)cos(15°). Use the sine-cosine product identity.
Substitute A = 75° and B = 15°:
sin(75°)cos(15°) = 1/2[sin(90°) + sin(60°)]
= 1/2[1 + √3 ÷ 2] = 1/2 + √3 ÷ 4
Example 2: Convert a Cosine Product
Simplify cos(75°)cos(15°).
Use the cosine-cosine identity:
cos(75°)cos(15°) = 1/2[cos(90°) + cos(60°)]
= 1/2[0 + 1/2] = 1/4
Sum-to-Product Identities
Example 3: Convert a Sum to a Product
Simplify sin(75°) + sin(15°).
Use the sine sum-to-product identity:
sin(75°) + sin(15°) = 2sin(45°)cos(30°)
= 2(√2 ÷ 2)(√3 ÷ 2) = √6 ÷ 2
Memory Tip
For sum-to-product formulas, the average angle comes first: calculate (A + B) ÷ 2. The difference angle follows: calculate (A − B) ÷ 2. Remember average, then difference.
Choosing the Correct Identity
| Expression type | Use | Main action |
|---|---|---|
| sin(A)cos(B) | Product-to-sum | Add and subtract the angles. |
| sin(A)sin(B) | Product-to-sum | Use cosine difference minus cosine sum. |
| sin(A) + sin(B) | Sum-to-product | Use average and difference angles. |
| cos(A) − cos(B) | Sum-to-product | Watch the negative sign. |
On a phone, swipe horizontally inside the table if needed.
Common Mistakes to Avoid
Mixing the two families: Products become sums; sums become products.
Forgetting the one-half: Every product-to-sum identity has a factor of 1/2.
Reversing the difference: Keep the order A − B consistent with the identity you choose.
Missing the negative sign: The cosine difference identity begins with −2.
Practice Problems
1. Rewrite cos(60°)cos(20°) as a sum.
2. Rewrite sin(70°)sin(10°) as a sum.
3. Rewrite cos(80°) + cos(40°) as a product.
4. Rewrite sin(50°) − sin(10°) as a product.
Tap to reveal the answers
1. 1/2[cos(80°) + cos(40°)]. 2. 1/2[cos(60°) − cos(80°)]. 3. 2cos(60°)cos(20°). 4. 2cos(30°)sin(20°).
Final Takeaway
Product-to-sum identities break products into sums. Sum-to-product identities combine sums into products. Identify the expression type, calculate the sum and difference angles carefully, and check all signs before simplifying.
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