Inverse function identities · derivation ·proof
Where tan(arcsin x) = x / √(1 − x²) comes from
Read the triangle. Build the triangle from sin θ = x/1 and take the ratio the tangent asks for.
Building it step by step
- 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.
Where this route is used
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}.
- 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 |
|---|---|---|---|
| -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 |
Neighbouring derivations
- cos(arcsin x) = √(1 − x²) — Cosine of an inverse sine
- sin(arcsin x) = x — Sine of an inverse sine
- tan θ = sin θ / cos θ — Quotient identity (tangent)
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from tan(arcsin(x)) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the tangent of an inverse sine page.