cosec⁻¹ and sec⁻¹

cosec x = 1/sin x is defined for x ≠ nπ and its values satisfy |y| ≥ 1, so its range is R − (−1, 1). On [−π/2, π/2] − {0} it is one-one and onto that range.

So cosec⁻¹ : R − (−1, 1) → [−π/2, π/2] − {0}. The angle 0 is removed because cosec 0 does not exist.

sec x = 1/cos x is defined for x ≠ (2n + 1)π/2, with range R − (−1, 1). On [0, π] − {π/2} it is one-one and onto that range.

So sec⁻¹ : R − (−1, 1) → [0, π] − {π/2}. The angle π/2 is removed because sec π/2 does not exist.

Both inverses reject inputs strictly between −1 and 1: cosec⁻¹(1/2) and sec⁻¹ 0 are undefined.

They follow the pattern of their partners: cosec⁻¹ 2 = π/6 and cosec⁻¹(−√2) = −π/4, as for sin⁻¹; sec⁻¹ 2 = π/3 and sec⁻¹(−2) = 2π/3, as for cos⁻¹.

Taking reciprocals converts one to the other: cosec⁻¹ x = sin⁻¹(1/x) and sec⁻¹ x = cos⁻¹(1/x) for |x| ≥ 1.