Product-to-sum identities
Rewrite a product of trigonometric functions as a sum.
Product-to-sum identities is one of 5 identities formulas in the trigonometry section of this library, and it is used at ap · university level.
Why product-to-sum identities works
Write out sin(A + B) and sin(A − B) side by side and add them. The two cos A sin B terms carry opposite signs and cancel, while the two sin A cos B terms are identical and double up. What is left is exactly the identity, so the rule is really the two sine expansions stacked on top of each other.
What each symbol means
$A,B$ are angle expressions.
Product-to-sum identities: when it holds
Use matching angle units throughout; other product-to-sum forms have different signs.
When it stops applying
This exact form only covers a sine times a cosine. The other pairings follow different patterns: 2 cos A cos B = cos(A − B) + cos(A + B), and 2 sin A sin B = cos(A − B) − cos(A + B), which subtracts and produces cosines rather than sines.
Product-to-sum identities: a worked example
$2\sin3x\cos x=\sin4x+\sin2x$.
The mistake to avoid
What people do: Students drop the leading 2 and write sin A cos B = sin(A + B) + sin(A − B).
Why it goes wrong: That doubles the true value. At A = 50 degrees and B = 20 degrees the right side is 1.4397, while sin 50 cos 20 is only 0.7198.
Do this instead: Keep the 2 attached, or divide the sum by 2 when you isolate the product: sin A cos B = ½[sin(A + B) + sin(A − B)]. Checking one numeric case catches this instantly.
Product-to-sum identities: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A,B$ are angle expressions.
- Check the conditions before substituting. Use matching angle units throughout; other product-to-sum forms have different signs.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Identities
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where product-to-sum identities comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Trigonometric Identities — the lesson behind this formula: simplify and prove relationships between trig functions.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about product-to-sum identities
Why would I want a sum instead of a product?
Because sums are far easier to integrate and to graph. In calculus and in signal work, sin 3x cos x is awkward, but the equivalent ½(sin 4x + sin 2x) integrates term by term.
Which angle keeps the sine, A or B?
The angle attached to the sine on the left stays first inside the results. Since sin(A − B) is not the same as sin(B − A), swapping them flips the sign of the second term and breaks the identity.
Is there a rule that goes the other direction?
Yes, the sum-to-product identities, such as sin P + sin Q = 2 sin((P + Q)/2) cos((P − Q)/2). They are these same statements rearranged, with the half-sum and half-difference of the angles playing the roles of A and B.
What happens if A and B are the same angle?
The difference term becomes sin 0 = 0, and you are left with 2 sin A cos A = sin 2A. So the double-angle identity falls out of this rule as the special case where the two angles match.