Inverse function identities · proof ·what it is and when to use it
Proof: cos(arcsin x) = √(1 − x²)
Cosine of an inverse sine — proved by pythagorean identity plus a range check. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
Two ingredients: the Pythagorean identity supplies the size, and the range of arcsin decides the sign. Skipping the second is the usual error.
- Let θ = arcsin xSo sin θ = x with θ in [−π/2, π/2].
- cos²θ = 1 − sin²θ = 1 − x²Pythagorean identity. · Pythagorean identity
- cos θ = +√(1 − x²)cosine is ≥ 0 on [−π/2, π/2].
- cos(arcsin x) = √(1 − x²)Substitute θ back.
At x = ±1 the cosine is 0 — the formula gives it, which is a cheap way to check you did not lose a factor.
Where the proof stops applying
x is a number from −1 to 1 — arcsin and arccos are only defined for inputs in this interval — outside it there is no angle to take.
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 |
|---|---|---|---|
| -1 | 0 | 0 | yes |
| -0.8 | 0.6 | 0.6 | yes |
| -0.6 | 0.8 | 0.8 | yes |
| -0.4 | 0.916515 | 0.916515 | yes |
| -0.2 | 0.979796 | 0.979796 | yes |
| 0 | 1 | 1 | yes |
Related
- How would I find cos(arcsin x) = √(1 − x²) myself? — the derivation, which is a different question from the proof.
- Cosine of an inverse sine: when to use it — the practical side.
- All inverse function identities · proof index