Sum and difference identities · trigonometric identity
tan A + tan B = sin(A + B) / (cos A cos B) — Tangent sum as a single fraction
When to use it
When a sum of two tangents has to become one fraction — standard in verification exercises, because the right side collapses two terms into one.
Why it is true
Write both tangents over their own cosines, take the common denominator cos A cos B, and the numerator is exactly the sine sum formula.
The full line-by-line version is on the proof page for tangent sum as a single fraction; the “how would I find this myself” version is in the derivation.
Where it comes from
Ask what the numerator wants to be. If a verification problem hands you tan A + tan B and asks for a single sine over a product of cosines, the middle step is always the common denominator.
- Target numerator: sin(A + B)What the exercise asks for.
- Expand the target: sin A cos B + cos A sin BThat is exactly what a common denominator produces. · Sine of a sum
- So the route is: quotient, then common denominatorConclusion.
Worked examples
- Verify tan A + tan B = sin(A + B)/(cos A cos B)Common denominator, then recognise the sine sum.
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° | 0.357929 | 0.357929 | yes |
| 15° | 0.564163 | 0.564163 | yes |
| 18° | 0.649839 | 0.649839 | yes |
| 22.5° | 0.783133 | 0.783133 | yes |
| 30° | 1.022579 | 1.022579 | yes |
| 37° | 1.274121 | 1.274121 | yes |
| 45° | 1.612801 | 1.612801 | yes |
| 53° | 2.040338 | 2.040338 | yes |
| 60° | 2.541835 | 2.541835 | yes |
| 67.5° | 3.338604 | 3.338604 | yes |
| 75° | 4.785831 | 4.785831 | yes |
| 120° | 0.873038 | 0.873038 | yes |
| 135° | 3.1653 | 3.1653 | yes |
| 150° | 8.937014 | 8.937014 | yes |
| 180° | -6.313752 | -6.313752 | yes |
| 210° | -1.668687 | -1.668687 | yes |
| 240° | 0.497154 | 0.497154 | yes |
| 300° | -2.115915 | -2.115915 | yes |
| 330° | -0.682455 | -0.682455 | yes |
| 360° | 0.158384 | 0.158384 | yes |
The mistake students make
Reversing it blindly: the identity needs both cosines non-zero, and it silently fails at A or B = 90°.
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 sumdetailsproof
- tan θ = sin θ / cos θQuotient identity (tangent)detailsproof
- sin(A + B) = sin A cos B + cos A sin BSine of a sumdetailsproof
Category hub: Sum and difference identities · all identities: /identities