Double-angle identities · proof ·what it is and when to use it
Proof: tan 2x = 2 tan x / (1 − tan²x)
Double-angle identity for tangent — proved by set b = a in the tangent sum. Every line below says which rule it uses, so nothing has to be taken on faith.
tan 2x = 2 tan x / (1 − tan²x)
The proof
One substitution in the tangent-of-a-sum identity.
- tan(A + B) = (tan A + tan B)/(1 − tan A tan B)Tangent of a sum. · Tangent of a sum
- B = A = xSpecialise.
- tan 2x = (tan x + tan x)/(1 − tan x·tan x)Substitute.
- tan 2x = 2 tan x/(1 − tan²x)Simplify both parts.
Where the proof stops applying
Both sides are undefined at the same angles — where a denominator in the proof reaches zero (for instance cos x = 0 in a tangent or secant form). Elsewhere the argument above holds for every real angle.
The same claim, checked numerically
A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).
| 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 |
Related
- How would I find tan 2x = 2 tan x / (1 − tan²x) myself? — the derivation, which is a different question from the proof.
- Double-angle identity for tangent: when to use it — the practical side.
- All double-angle identities · proof index