Inverse function identities · proof ·what it is and when to use it
Proof: tan(arcsin x) = x / √(1 − x²)
Tangent of an inverse sine — proved by quotient identity applied to the pair. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
One line, if the two ingredients are already allowed: tangent is sine over cosine.
- Let θ = arcsin xSo sin θ = x, cos θ = √(1 − x²). · Cosine of an inverse sine
- tan θ = sin θ / cos θQuotient identity. · Quotient identity (tangent)
- tan(arcsin x) = x / √(1 − x²)Substitute both.
The sign of the result follows x, which is right: arcsin is odd and tangent is odd, so an odd composed with an odd is odd.
Where the proof stops applying
x is a number from −1 to 1 — and not ±1: at those two values the angle is ±π/2 and the tangent does not exist.
Outside that interval the inverse function returns nothing at all, so the statement is not false there — it has no meaning. The proof above never leaves the interval.
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 |
|---|---|---|---|
| -0.8 | -1.333333 | -1.333333 | yes |
| -0.6 | -0.75 | -0.75 | yes |
| -0.4 | -0.436436 | -0.436436 | yes |
| -0.2 | -0.204124 | -0.204124 | yes |
| 0 | 0 | 0 | yes |
| 0.2 | 0.204124 | 0.204124 | yes |
Related
- How would I find tan(arcsin x) = x / √(1 − x²) myself? — the derivation, which is a different question from the proof.
- Tangent of an inverse sine: when to use it — the practical side.
- All inverse function identities · proof index