Sum and difference identities · trigonometric identity

sin(A − B) = sin A cos B − cos A sin B — Sine of a difference

When to use it

Same family with a minus. It is the sum formula with B replaced by −B, so the second term changes sign — handy for exact values like sin 15° = sin(45° − 30°).

Why it is true

Put −B in place of B in the sum formula and use that sine is odd while cosine is even: the first term keeps its sign, the second flips.

The full line-by-line version is on the proof page for sine of a difference; the “how would I find this myself” version is in the derivation.

Where it comes from

Remember one formula, derive the other. There is no need to memorise the sum and difference versions separately — the sign of the middle term follows from the sign inside the angle.

1. Sum version has '+' on both sides
   sin(A + B) = … + …
2. Difference version flips the second term
   Replace B by −B.
3. Mnemonic: sine keeps the sign
   sin(A ± B) = … ± … mirrors the sign inside.

Worked examples

• Find the exact value of sin 15°
  (√6 − √2)/4

Checked at these angles

Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.

The mistake students make

Try it

• Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
• Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
• Cheat sheet entry for sum and difference identities.

Related identities

• sin(A + B) = sin A cos B + cos A sin BSine of a sumdetailsproof
• cos(A − B) = cos A cos B + sin A sin BCosine of a differencedetailsproof
• sin 2x = 2 sin x cos xDouble-angle identity for sinedetailsproof

Category hub: Sum and difference identities · all identities: /identities

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7.2–7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error