Inverse function identities · trigonometric identity
cos(arccos x) = x — Cosine of an inverse cosine
When to use it
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.
Why it is true
arccos x is defined as the angle in [0, π] whose cosine is x, so feeding that angle back through cosine returns x — the definition read forwards.
The full line-by-line version is on the proof page for cosine of an inverse cosine; the “how would I find this myself” version is in the derivation.
Where it 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.
- 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.
Worked examples
No worked example is written for this identity yet —the examples index lists what is covered, and the pattern below shows the identity working at real angles.
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Value of x | 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 |
| 0.2 | 0.2 | 0.2 | yes |
| 0.4 | 0.4 | 0.4 | yes |
| 0.6 | 0.6 | 0.6 | yes |
| 0.8 | 0.8 | 0.8 | yes |
| 1 | 1 | 1 | yes |
What x is on this page
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.
This is the one family on the site where the letter is not an angle. Every other page samples angles across [−4π, 4π]; here the sampler stays inside the interval above, because outside it the inverse function has no value to compare.
The mistake students make
Assuming arccos and arcsin share the same output range. arcsin answers in [−π/2, π/2]; arccos answers in [0, π]. That difference is why arccos has no odd/even symmetry.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for inverse function identities.
Related identities
- sin(arcsin x) = xSine of an inverse sinedetailsproof
- sin(arccos x) = √(1 − x²)Sine of an inverse cosinedetailsproof
- arccos(−x) = π − arccos xInverse cosine of a negativedetailsproof
Category hub: Inverse function identities · all identities: /identities