Inverse function identities · trigonometric identity
tan(arcsin x) = x / √(1 − x²) — Tangent of an inverse sine
When to use it
When a substitution like t = √(1 − x²) or x = sin θ has to be carried through a tangent: this is the quotient of the two previous identities, and the form you need in integrals such as ∫ dx/(1 − x²)^{3/2}.
Why it is true
tan θ = sin θ / cos θ. With θ = arcsin x the numerator is x and the denominator is √(1 − x²), so the ratio follows directly.
The full line-by-line version is on the proof page for tangent of an inverse sine; the “how would I find this myself” version is in the derivation.
Where it comes from
Read the triangle. Build the triangle from sin θ = x/1 and take the ratio the tangent asks for.
- Opposite = x, hypotenuse = 1, adjacent = √(1 − x²)Pythagoras.
- tan θ = opposite / adjacentDefinition of tangent.
- tan(arcsin x) = x / √(1 − x²)Put the sides in.
Triangle derivations are fast but assume x > 0; the algebraic proof is what carries the negative half of the domain.
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 |
|---|---|---|---|
| -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 |
| 0.4 | 0.436436 | 0.436436 | yes |
| 0.6 | 0.75 | 0.75 | yes |
| 0.8 | 1.333333 | 1.333333 | yes |
What x is on this page
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.
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
Forgetting that x = ±1 is excluded. There the cosine is 0, so the tangent does not exist — the formula's denominator says so, and so does the angle ±π/2.
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
- cos(arcsin x) = √(1 − x²)Cosine of an inverse sinedetailsproof
- sin(arcsin x) = xSine of an inverse sinedetailsproof
- tan θ = sin θ / cos θQuotient identity (tangent)detailsproof
Category hub: Inverse function identities · all identities: /identities