Double-angle identities · trigonometric identity
tan 2x = 2 tan x / (1 − tan²x) — Double-angle identity for tangent
When to use it
Everything in the problem is a tangent — solving tan 2x = 1, or rewriting a slope-doubling situation. Also the formula that shows why tan has period 180° but tan 2x compresses it.
Why it is true
Set B = A in the tangent sum formula: the numerator becomes 2 tan A, the denominator 1 − tan²A.
The full line-by-line version is on the proof page for double-angle identity for tangent; the “how would I find this myself” version is in the derivation.
Where it comes from
Or divide the other two. If the tangent formula is not memorised: tan 2x = sin 2x / cos 2x, then substitute the sine and cosine double-angle forms and divide top and bottom by cos²x.
- tan 2x = sin 2x/cos 2xQuotient identity. · Quotient identity (tangent)
- = 2 sin x cos x/(cos²x − sin²x)Both double-angle forms. · Double-angle identity for sine, Double-angle identity for cosine (form 1)
- divide by cos²xTurns each piece into a tangent.
- = 2 tan x/(1 − tan²x)Result.
Worked examples
- Solve tan 2x = 1 on [0, π)x = π/8 and x = 5π/8
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 | 0 | yes |
| 7.5° | 0.267949 | 0.267949 | yes |
| 15° | 0.57735 | 0.57735 | yes |
| 18° | 0.726543 | 0.726543 | yes |
| 22.5° | 1 | 1 | yes |
| 30° | 1.732051 | 1.732051 | yes |
| 37° | 3.487414 | 3.487414 | yes |
| 53° | -3.487414 | -3.487414 | yes |
| 60° | -1.732051 | -1.732051 | yes |
| 67.5° | -1 | -1 | yes |
| 75° | -0.57735 | -0.57735 | yes |
| 120° | 1.732051 | 1.732051 | yes |
| 150° | -1.732051 | -1.732051 | yes |
| 180° | -0 | -0 | yes |
| 210° | 1.732051 | 1.732051 | yes |
| 240° | -1.732051 | -1.732051 | yes |
| 300° | 1.732051 | 1.732051 | yes |
| 330° | -1.732051 | -1.732051 | yes |
| 360° | -0 | -0 | yes |
The mistake students make
Undefined points: the right side blows up when tan²x = 1, i.e. x = 45°, which is exactly where 2x = 90° and tan 2x genuinely has no value.
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 double-angle identities.
Related identities
- tan(A + B) = (tan A + tan B) / (1 − tan A tan B)Tangent of a sumdetailsproof
- sin 2x = 2 sin x cos xDouble-angle identity for sinedetailsproof
- cos 2x = cos²x − sin²xDouble-angle identity for cosine (form 1)detailsproof
- 1 + tan²θ = sec²θPythagorean identity (tangent form)detailsproof
Category hub: Double-angle identities · all identities: /identities