Sum and difference identities · trigonometric identity
cos(A + B) = cos A cos B − sin A sin B — Cosine of a sum
When to use it
Expanding a cosine of a sum — and the reason cos 2A and the power-reducing formulas look the way they do. The sign here is minus, which surprises most students the first time.
Why it is true
Same rotation as the sine version, but read the horizontal coordinate: x cos B − y sin B with (x, y) = (cos A, sin A).
The full line-by-line version is on the proof page for cosine of a sum; the “how would I find this myself” version is in the derivation.
Where it comes from
Derive the double-angle formula from it. Setting B = A is the standard route to cos 2A, and from there to the three cosine double-angle forms.
- cos(A + B) = cos A cos B − sin A sin BStart. · Cosine of a sum
- B = ASpecialise.
- cos 2A = cos²A − sin²AMultiply matching functions. · Double-angle identity for cosine (form 1)
Worked examples
- Derive the double-angle formulas from the sum formulasSet B = A in each sum formula; cosine then yields two further forms via sin² + cos² = 1.
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.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 0.987688 | 0.987688 | yes |
| 7.5° | 0.938191 | 0.938191 | yes |
| 15° | 0.85264 | 0.85264 | yes |
| 18° | 0.809017 | 0.809017 | yes |
| 22.5° | 0.734323 | 0.734323 | yes |
| 30° | 0.587785 | 0.587785 | yes |
| 37° | 0.430511 | 0.430511 | yes |
| 45° | 0.233445 | 0.233445 | yes |
| 53° | 0.026177 | 0.026177 | yes |
| 60° | -0.156434 | -0.156434 | yes |
| 67.5° | -0.346117 | -0.346117 | yes |
| 75° | -0.522499 | -0.522499 | yes |
| 90° | -0.809017 | -0.809017 | yes |
| 120° | -0.987688 | -0.987688 | yes |
| 135° | -0.85264 | -0.85264 | yes |
| 150° | -0.587785 | -0.587785 | yes |
| 180° | 0.156434 | 0.156434 | yes |
| 210° | 0.809017 | 0.809017 | yes |
| 240° | 0.987688 | 0.987688 | yes |
| 270° | 0.587785 | 0.587785 | yes |
| 300° | -0.156434 | -0.156434 | yes |
| 330° | -0.809017 | -0.809017 | yes |
| 360° | -0.987688 | -0.987688 | yes |
The mistake students make
Keeping the plus sign. cos(A + B) has a minus in the middle, while sin(A + B) has a plus. Sine 'copies' the sign; cosine 'opposes' it.
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
- cos(A − B) = cos A cos B + sin A sin BCosine of a differencedetailsproof
- sin(A + B) = sin A cos B + cos A sin BSine of a sumdetailsproof
- cos 2x = cos²x − sin²xDouble-angle identity for cosine (form 1)detailsproof
- cos²x = (1 + cos 2x)/2Power-reducing formula for cos²detailsproof
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