Finding principal values

The principal value is the value of the inverse function that lies in its principal range. The method: set the expression equal to y, rewrite as a trigonometric equation, and pick the one solution inside the principal range.

sin⁻¹(1/√2): sin y = 1/√2 with y ∈ [−π/2, π/2] gives y = π/4.

cot⁻¹(−1/√3): cot y = −1/√3 = −cot(π/3) = cot(π − π/3), and 2π/3 lies in (0, π), so the principal value is 2π/3.

cos⁻¹(√3/2) = π/6, tan⁻¹(−√3) = −π/3, cosec⁻¹ 2 = π/6, sec⁻¹(2/√3) = π/6.

Sums are found term by term: tan⁻¹ 1 + cos⁻¹(−1/2) + sin⁻¹(−1/2) = π/4 + 2π/3 − π/6 = 3π/4.

tan⁻¹ √3 − sec⁻¹(−2) = π/3 − 2π/3 = −π/3. A negative total is fine; only each individual term must sit in its own principal range.

Check the answer by plugging back: the angle must give the right trigonometric value and lie in the range. 4π/3 has cot 4π/3 = 1/√3 but lies outside (0, π), so it is never a value of cot⁻¹.