Simplest form and identities

Many inverse trig expressions simplify with a clever substitution: replace x with a trig function, so that a square root or a fraction turns into a single trig ratio.

The square root tells you which substitution to pick, because of the identities 1 – sin²θ = cos²θ, 1 + tan²θ = sec²θ and sec²θ – 1 = tan²θ:

Then use a double- or triple-angle formula, and finally cancel the inverse.

Example: tan⁻¹(2x/(1 – x²)) for –1 < x < 1. Put x = tan θ. Then 2tan θ/(1 – tan²θ) = tan 2θ, so the expression is tan⁻¹(tan 2θ) = 2θ = 2 tan⁻¹x.

Why the condition –1 < x < 1? It keeps θ between –π/4 and π/4, so 2θ stays inside (–π/2, π/2) and the cancelling is allowed. Always check the new angle lies in the branch before cancelling.

Example: cos⁻¹(√(1 – x²)) for 0 ≤ x ≤ 1. Put x = sin θ: √(1 – sin²θ) = cos θ, so this is cos⁻¹(cos θ) = θ = sin⁻¹x.

To prove a sum of two inverses equals a third, change each term to tan⁻¹ and use tan(A + B) = (tan A + tan B)/(1 – tan A tan B). A right triangle helps convert: sin⁻¹(3/5) is the angle in a 3-4-5 triangle with opposite 3 and hypotenuse 5, so its tangent is 3/4 and sin⁻¹(3/5) = tan⁻¹(3/4).