INVERSE TRIGO

Domain, Range, and Monotonicity of y = sin⁻¹(x)

• Domain: [-1, 1]\n- Principal Value Branch (Range): [-π/2, π/2]\n- Monotonicity: Strictly Increasing in [-1, 1]\n- Parity: Odd Function [sin⁻¹(-x) = -sin⁻¹(x)]

Domain, Range, and Monotonicity of y = cos⁻¹(x)

• Domain: [-1, 1]\n- Principal Value Branch (Range): [0, π]\n- Monotonicity: Strictly Decreasing in [-1, 1]\n- Parity: Neither Even nor Odd [cos⁻¹(-x) = π - cos⁻¹(x)]

Domain, Range, and Monotonicity of y = tan⁻¹(x)

• Domain: ℝ (All Real Numbers)\n- Principal Value Branch (Range): (-π/2, π/2)\n- Monotonicity: Strictly Increasing in (-∞, ∞)\n- Parity: Odd Function [tan⁻¹(-x) = -tan⁻¹(x)]\n- Asymptotes: y = π/2 and y = -π/2

Domain, Range, and Monotonicity of y = cot⁻¹(x)

• Domain: ℝ (All Real Numbers)\n- Principal Value Branch (Range): (0, π)\n- Monotonicity: Strictly Decreasing in (-∞, ∞)\n- Parity: Neither Even nor Odd [cot⁻¹(-x) = π - cot⁻¹(x)]\n- Asymptotes: y = 0 and y = π

Domain, Range, and Monotonicity of y = sec⁻¹(x)

• Domain: (-∞, -1] ∪ [1, ∞) [or ℝ - (-1, 1)]\n- Principal Value Branch (Range): [0, π] - {π/2}\n- Monotonicity: Strictly Increasing in (-∞, -1] and [1, ∞)\n- Parity: Neither Even nor Odd [sec⁻¹(-x) = π - sec⁻¹(x)]\n- Asymptote: y = π/2

Domain, Range, and Monotonicity of y = cosec⁻¹(x)

• Domain: (-∞, -1] ∪ [1, ∞) [or ℝ - (-1, 1)]\n- Principal Value Branch (Range): [-π/2, π/2] - {0}\n- Monotonicity: Strictly Decreasing in (-∞, -1] and [1, ∞)\n- Parity: Odd Function [cosec⁻¹(-x) = -cosec⁻¹(x)]\n- Asymptote: y = 0

Negative Argument Properties (Odd / Neither-Odd Identities)

1. sin⁻¹(-x) = -sin⁻¹(x), x ∈ [-1, 1]\n2. tan⁻¹(-x) = -tan⁻¹(x), x ∈ ℝ\n3. cosec⁻¹(-x) = -cosec⁻¹(x), |x| ≥ 1\n4. cos⁻¹(-x) = π - cos⁻¹(x), x ∈ [-1, 1]\n5. cot⁻¹(-x) = π - cot⁻¹(x), x ∈ ℝ\n6. sec⁻¹(-x) = π - sec⁻¹(x), |x| ≥ 1

Complementary Angle Identities for ITF

• sin⁻¹(x) + cos⁻¹(x) = π/2, for x ∈ [-1, 1]\n- tan⁻¹(x) + cot⁻¹(x) = π/2, for x ∈ ℝ\n- sec⁻¹(x) + cosec⁻¹(x) = π/2, for |x| ≥ 1

Reciprocal Identities for ITF

• sin⁻¹(1/x) = cosec⁻¹(x), for |x| ≥ 1\n- cos⁻¹(1/x) = sec⁻¹(x), for |x| ≥ 1\n- tan⁻¹(1/x) = cot⁻¹(x), for x > 0\n- tan⁻¹(1/x) = -π + cot⁻¹(x), for x < 0

Addition Formula for tan⁻¹(x) + tan⁻¹(y)

• If xy < 1: tan⁻¹(x) + tan⁻¹(y) = tan⁻¹((x + y) / (1 - xy))\n- If xy > 1 and x > 0, y > 0: tan⁻¹(x) + tan⁻¹(y) = π + tan⁻¹((x + y) / (1 - xy))\n- If xy > 1 and x < 0, y < 0: tan⁻¹(x) + tan⁻¹(y) = -π + tan⁻¹((x + y) / (1 - xy))

Subtraction Formula for tan⁻¹(x) - tan⁻¹(y)

• If xy > -1: tan⁻¹(x) - tan⁻¹(y) = tan⁻¹((x - y) / (1 + xy))\n- If xy < -1 and x > 0, y < 0: tan⁻¹(x) - tan⁻¹(y) = π + tan⁻¹((x - y) / (1 + xy))\n- If xy < -1 and x < 0, y > 0: tan⁻¹(x) - tan⁻¹(y) = -π + tan⁻¹((x - y) / (1 + xy))

Addition and Subtraction Formulas for sin⁻¹(x) ± sin⁻¹(y)

• sin⁻¹(x) + sin⁻¹(y) = sin⁻¹(x√(1 - y²) + y√(1 - x²)), for x,y ≥ 0 and x² + y² ≤ 1\n- sin⁻¹(x) - sin⁻¹(y) = sin⁻¹(x√(1 - y²) - y√(1 - x²)), for x,y ≥ 0

Addition and Subtraction Formulas for cos⁻¹(x) ± cos⁻¹(y)

• cos⁻¹(x) + cos⁻¹(y) = cos⁻¹(xy - √((1 - x²)(1 - y²))), for x,y ∈ [-1, 1] and x + y ≥ 0\n- cos⁻¹(x) - cos⁻¹(y) = cos⁻¹(xy + √((1 - x²)(1 - y²))), for x,y ∈ [-1, 1] and x ≤ y

Multiple Angle Formulas for 2 tan⁻¹(x)

• In terms of tan⁻¹: 2 tan⁻¹(x) = tan⁻¹(2x / (1 - x²)), for |x| < 1\n- In terms of sin⁻¹: 2 tan⁻¹(x) = sin⁻¹(2x / (1 + x²)), for |x| ≤ 1\n- In terms of cos⁻¹: 2 tan⁻¹(x) = cos⁻¹((1 - x²) / (1 + x²)), for x ≥ 0

Multiple Angle Formulas for 2 sin⁻¹(x) and 2 cos⁻¹(x)

• 2 sin⁻¹(x) = sin⁻¹(2x√(1 - x²)), for -1/√2 ≤ x ≤ 1/√2\n- 2 cos⁻¹(x) = cos⁻¹(2x² - 1), for 0 ≤ x ≤ 1

Triple Angle Formulas for 3 sin⁻¹(x), 3 cos⁻¹(x), and 3 tan⁻¹(x)

• 3 sin⁻¹(x) = sin⁻¹(3x - 4x³), for x ∈ [-1/2, 1/2]\n- 3 cos⁻¹(x) = cos⁻¹(4x³ - 3x), for x ∈ [1/2, 1]\n- 3 tan⁻¹(x) = tan⁻¹((3x - x³) / (1 - 3x²)), for x ∈ (-1/√3, 1/√3)

Conversion / Cancellation Identities f(f⁻¹(x)) vs. f⁻¹(f(x))

1. f(f⁻¹(x)) = x for x in the DOMAIN of f⁻¹. (e.g., sin(sin⁻¹(x)) = x for x ∈ [-1, 1])\n2. f⁻¹(f(x)) = x ONLY if x lies inside the PRINCIPAL VALUE BRANCH (RANGE) of f⁻¹.\n- Example: sin⁻¹(sin(x)) = x only if x ∈ [-π/2, π/2].\n- If x is outside the range, adjust x using periodicity (e.g., sin⁻¹(sin(2π/3)) = π/3).