Inverse function identities · derivation ·proof
Where sin(arccos x) = √(1 − x²) comes from
Transfer it through the complementary-angle identity. If you already trust arcsin x + arccos x = π/2, this falls out without any new algebra.
Building it step by step
- arccos x = π/2 − arcsin xComplementary-angle identity. · Inverse sine plus inverse cosine
- sin(arccos x) = sin(π/2 − arcsin x)Substitute.
- = cos(arcsin x) = √(1 − x²)Cofunction, then the known cosine form. · Cofunction identity for sine, Cosine of an inverse sine
This is the cheapest way to remember the pair: one of them, plus the complementary-angle identity, gives the other.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- arcsin x + arccos x = π/2 — Inverse sine plus inverse cosine
- sin θ = cos(90° − θ) — Cofunction identity for sine
- cos(arcsin x) = √(1 − x²) — Cosine of an inverse sine
That list is the argument for learning fewer formulas: most of this catalogue is a short chain away from the unit circle and the sum formulas.
Where this route is used
Same shape as cos(arcsin x) and the reason both are written without a ±: the angle arccos returns lives in [0, π], where sine is never negative either.
- simplify
- integrate
The result, checked at real values
Deriving a formula and testing it are different things; this table is the test. The numbers come from the same engine as the verifier, computed when the site was built.
| 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 |
Neighbouring derivations
- cos(arcsin x) = √(1 − x²) — Cosine of an inverse sine
- arcsin x + arccos x = π/2 — Inverse sine plus inverse cosine
- cos(arccos x) = x — Cosine of an inverse cosine
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from arcsin x + arccos x = π/2 and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the sine of an inverse cosine page.