Inverse function identities · derivation ·proof
Where cos(arccos x) = x comes from
From the restricted cosine curve. Cosine is not one-to-one on the whole line, so an inverse only exists after a restriction. Following that restriction gives both the identity and its domain.
Building it step by step
- cos θ = x, θ chosen in [0, π]The branch where cosine never repeats.
- θ = arccos xNaming the branch's solution.
- cos(arccos x) = xSubstitute back.
Why [0, π] and not [−π/2, π/2]? Because cosine is monotone on [0, π] — it runs from 1 down to −1 exactly once.
Where this route is used
The cosine twin of sin(arcsin x) = x: an inverse cosine wrapped in cosine returns its input. Useful when clearing a trig function off a variable in an equation.
- simplify
- verify
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 | -1 | -1 | yes |
| -0.8 | -0.8 | -0.8 | yes |
| -0.6 | -0.6 | -0.6 | yes |
| -0.4 | -0.4 | -0.4 | yes |
| -0.2 | -0.2 | -0.2 | yes |
| 0 | 0 | 0 | yes |
Neighbouring derivations
- sin(arcsin x) = x — Sine of an inverse sine
- sin(arccos x) = √(1 − x²) — Sine of an inverse cosine
- arccos(−x) = π − arccos x — Inverse cosine of a negative
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from cos(arccos(x)) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the cosine of an inverse cosine page.