Inverse Trig Function Properties: Identities & Graphs

Inverse trigonometric functions become far more useful once their identities are known, because the identities let you rewrite sums, doubles, and negatives of these functions into simpler single terms. Instead of leaving an answer as a messy expression like sin⁻¹x + cos⁻¹x, one identity collapses it instantly to π/2.

These identities matter in calculus, where integrals of rational functions are often evaluated through arctangent substitutions, and in physics, where two angle measurements must be combined into one clean result. Knowing the handful of key identities saves pages of algebra and prevents the quadrant mistakes that come from working with raw angles.

Key Concepts

• Complementary identity: pairs of inverse functions whose angles add to a right angle, like sin⁻¹x + cos⁻¹x = π/2.
• Double-angle identity: rewrites 2tan⁻¹x or 2sin⁻¹x as a single inverse trigonometric term.
• Domain condition: every identity below works only for inputs inside the stated interval; outside it the formula can fail or need an extra π.
• Principal value consistency: both sides of an identity must be read as principal values for the equality to hold.
• All angles in this article are in radians.

Formulas

• sin⁻¹x + cos⁻¹x = π/2, for −1 ≤ x ≤ 1.
• tan⁻¹x + cot⁻¹x = π/2, for every real x.
• sec⁻¹x + cosec⁻¹x = π/2, for |x| ≥ 1.
• sin(sin⁻¹x) = x, for −1 ≤ x ≤ 1. Note that sin⁻¹(sin x) = x only when −π/2 ≤ x ≤ π/2.
• sin⁻¹(−x) = −sin⁻¹x and tan⁻¹(−x) = −tan⁻¹x; but cos⁻¹(−x) = π − cos⁻¹x, for −1 ≤ x ≤ 1.
• 2tan⁻¹x = tan⁻¹(2x/(1 − x²)), for |x| < 1.
• 2sin⁻¹x = sin⁻¹(2x√(1 − x²)), for |x| ≤ 1/√2.
• tan⁻¹x + tan⁻¹y = tan⁻¹((x + y)/(1 − xy)), when xy < 1.

Worked Example

Simplify 2tan⁻¹(1/3) into a single arctangent.

1. Use 2tan⁻¹x = tan⁻¹(2x/(1 − x²)) with x = 1/3. First check the condition: |1/3| < 1, so the formula applies directly.
2. Compute the numerator: 2x = 2/3.
3. Compute the denominator: 1 − x² = 1 − 1/9 = 8/9.
4. Divide: (2/3)/(8/9) = (2/3) × (9/8) = 18/24 = 3/4.
5. Therefore 2tan⁻¹(1/3) = tan⁻¹(3/4).
6. Quick check with decimals: tan⁻¹(1/3) ≈ 18.43°, doubling gives ≈ 36.87°, and tan⁻¹(0.75) ≈ 36.87°. They match.

The simplified form is tan⁻¹(3/4).

Common Misconceptions

• "2tan⁻¹x = tan⁻¹(2x/(1 − x²)) works for every x." It does not. The formula needs |x| < 1; for x = 2, the left side is about 126.9° but the right side gives tan⁻¹(−4/3) ≈ −53.1°, which is wrong by 180°.
• "sin⁻¹x + cos⁻¹x = π/2 for any x." The identity requires −1 ≤ x ≤ 1, since outside that interval the inverse functions are not even defined over the reals.
• "tan⁻¹x + tan⁻¹y always equals tan⁻¹((x + y)/(1 − xy))." Only when xy < 1. When xy > 1 the true sum differs by π, because the arctangent sum can cross into the next quadrant.

Real-World Applications

• Calculus: integrals like the integral of dx/(1 + x²) evaluate to tan⁻¹x, and the addition identities combine the results of definite integrals cleanly.
• Optics: when light bends through several lenses, the total deviation angle is found by adding inverse trigonometric terms with these identities.
• Navigation: two bearing corrections expressed as arctangents are merged into one heading using the addition formula.
• Control systems: phase angles of electronic filters are sums of arctangents, simplified with the double-angle identities.

The three most-used identities are the complementary pair sin⁻¹x + cos⁻¹x = π/2, the cancellation sin(sin⁻¹x) = x, and the double-angle form 2tan⁻¹x = tan⁻¹(2x/(1 − x²)) valid for |x| < 1. Every one of them carries a domain condition, and checking that condition before applying the formula is what separates a correct simplification from a wrong one.