Inverse Trigonometric Functions

sin⁻¹(–x) = –sin⁻¹(x) , x ∈ [–1,1]
cos⁻¹(–x) = π – cos⁻¹(x) , x ∈ [–1,1]
tan⁻¹(–x) = –tan⁻¹(x) , x ∈ ℝ
cosec⁻¹(–x) = -cosec⁻¹(x) , x ∈ (-∞, -1] ∪ [1, ∞)
sec⁻¹(–x) = π - sec⁻¹(x) , x ∈ (-∞, -1] ∪ [1, ∞)
cot⁻¹(–x) = π - cot⁻¹(x) , x ∈ ℝ

sin⁻¹(1/x) = cosec⁻¹(x)
cos⁻¹(1/x) = sec⁻¹(x)
tan⁻¹(1/x) = cot⁻¹(x), x > 0

sin⁻¹(x) + cos⁻¹(x) = π/2
tan⁻¹(x) + cot⁻¹(x) = π/2
sec⁻¹(x) + cosec⁻¹(x) = π/2

tan⁻¹x + tan⁻¹y
= tan⁻¹[(x+y)/(1–xy)], xy < 1

sin⁻¹x ± sin⁻¹y = sin⁻¹[x√(1–y²) ± y√(1–x²)]
cos⁻¹x ± cos⁻¹y = cos⁻¹[x√(1–y²) ∓ y√(1–x²)]

2tan⁻¹x = sin⁻¹(2x/1+x²)
2tan⁻¹x = cos⁻¹(1–x²/1+x²)
2tan⁻¹x = tan⁻¹(2x/1–x²)

sin⁻¹: [–1,1] → [–π/2, π/2]
cos⁻¹: [–1,1] → [0, π]
tan⁻¹: ℝ → (–π/2, π/2)