Sum and difference identities · trigonometric identity
tan(A + B) = (tan A + tan B) / (1 − tan A tan B) — Tangent of a sum
When to use it
When the problem is written in tangents — slopes of two lines, angles between lines, or exact values like tan 75°. Divide the sine sum by the cosine sum and this is what falls out.
Why it is true
tan = sin/cos. Expand sin(A + B) and cos(A + B), then divide numerator and denominator by cos A cos B; every term becomes a tangent.
The full line-by-line version is on the proof page for tangent of a sum; the “how would I find this myself” version is in the derivation.
Where it comes from
Force tangents to appear. The trick is the division by cos A cos B — it is what turns mixed sine/cosine products into tangents.
- Every term should become tanGoal.
- tan = sin/cos needs a cos in the denominatorSo divide by the product cos A cos B.
- (sin A cos B)/(cos A cos B) = tan ACancellation, term by term.
Worked examples
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° | 0.158384 | 0.158384 | yes |
| 7.5° | 0.368919 | 0.368919 | yes |
| 15° | 0.612801 | 0.612801 | yes |
| 18° | 0.726543 | 0.726543 | yes |
| 22.5° | 0.92439 | 0.92439 | yes |
| 30° | 1.376382 | 1.376382 | yes |
| 37° | 2.096544 | 2.096544 | yes |
| 45° | 4.1653 | 4.1653 | yes |
| 53° | 38.188459 | 38.188459 | yes |
| 60° | -6.313752 | -6.313752 | yes |
| 67.5° | -2.710619 | -2.710619 | yes |
| 75° | -1.631852 | -1.631852 | yes |
| 120° | 0.158384 | 0.158384 | yes |
| 135° | 0.612801 | 0.612801 | yes |
| 150° | 1.376382 | 1.376382 | yes |
| 180° | -6.313752 | -6.313752 | yes |
| 210° | -0.726543 | -0.726543 | yes |
| 240° | 0.158384 | 0.158384 | yes |
| 300° | -6.313752 | -6.313752 | yes |
| 330° | -0.726543 | -0.726543 | yes |
| 360° | 0.158384 | 0.158384 | yes |
The mistake students make
Forgetting the denominator. tan(A + B) is not tan A + tan B. Also: the formula is undefined when A + B = 90°, or when the denominator 1 − tan A tan B is zero — same event, described twice.
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
- tan(A − B) = (tan A − tan B) / (1 + tan A tan B)Tangent of a differencedetailsproof
- tan A + tan B = sin(A + B) / (cos A cos B)Tangent sum as a single fractiondetailsproof
- tan 2x = 2 tan x / (1 − tan²x)Double-angle identity for tangentdetailsproof
- tan θ = sin θ / cos θQuotient identity (tangent)detailsproof
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