Identities JEE adds

The rationalised textbook proves only the undoing rules, but JEE questions use the standard identities below. Each follows from the principal ranges.

Negative inputs: sin⁻¹(−x) = −sin⁻¹ x, tan⁻¹(−x) = −tan⁻¹ x, cosec⁻¹(−x) = −cosec⁻¹ x; while cos⁻¹(−x) = π − cos⁻¹ x, cot⁻¹(−x) = π − cot⁻¹ x, sec⁻¹(−x) = π − sec⁻¹ x.

Complementary pairs: sin⁻¹ x + cos⁻¹ x = π/2 for x ∈ [−1, 1]; tan⁻¹ x + cot⁻¹ x = π/2 for all real x; sec⁻¹ x + cosec⁻¹ x = π/2 for |x| ≥ 1.

Why the first holds: if sin⁻¹ x = θ, then cos(π/2 − θ) = sin θ = x and π/2 − θ lies in [0, π], so cos⁻¹ x = π/2 − θ.

Reciprocals: cosec⁻¹ x = sin⁻¹(1/x) and sec⁻¹ x = cos⁻¹(1/x) for |x| ≥ 1; cot⁻¹ x = tan⁻¹(1/x) for x > 0 only. For x < 0, cot⁻¹ x = π + tan⁻¹(1/x).

Addition: tan⁻¹ x + tan⁻¹ y = tan⁻¹((x + y)/(1 − xy)) when xy < 1. Example: tan⁻¹(1/2) + tan⁻¹(1/3) = tan⁻¹((5/6)/(5/6)) = tan⁻¹ 1 = π/4.

Doubling: 2 tan⁻¹ x = tan⁻¹(2x/(1 − x²)) for |x| < 1, and 2 tan⁻¹ x = sin⁻¹(2x/(1 + x²)) for |x| ≤ 1.