Inverse Trigonometric Functions 

Definition of Inverse Trigonometric Functions

• sin⁻¹x = θ ⇔ sinθ = x
• cos⁻¹x = θ ⇔ cosθ = x
• tan⁻¹x = θ ⇔ tanθ = x
• cot⁻¹x = θ ⇔ cotθ = x
• sec⁻¹x = θ ⇔ secθ = x
• cosec⁻¹x = θ ⇔ cosecθ = x

Domain and Range (Principal Values)

• y = sin⁻¹x Domain: [-1, 1] Range: [−π/2, π/2]
• y = cos⁻¹x Domain: [-1, 1] Range: [0, π]
• y = tan⁻¹x Domain: (−∞, ∞) Range: (−π/2, π/2)
• y = cot⁻¹x Domain: (−∞, ∞) Range: (0, π)
• y = sec⁻¹x Domain: (−∞, −1] ∪ [1, ∞) Range: [0, π], y ≠ π/2
• y = cosec⁻¹x Domain: (−∞, −1] ∪ [1, ∞) Range: [−π/2, π/2], y ≠ 0

Basic Properties

• sin(sin⁻¹x) = x, −1 ≤ x ≤ 1
• cos(cos⁻¹x) = x, −1 ≤ x ≤ 1
• tan(tan⁻¹x) = x, x ∈ ℝ
• sin⁻¹(sinθ) = θ, only if −π/2 ≤ θ ≤ π/2
• cos⁻¹(cosθ) = θ, only if 0 ≤ θ ≤ π
• tan⁻¹(tanθ) = θ, only if −π/2 < θ < π/2

Negative Argument Properties

• sin⁻¹(−x) = −sin⁻¹x
• cos⁻¹(−x) = π − cos⁻¹x
• tan⁻¹(−x) = −tan⁻¹x
• cot⁻¹(−x) = π − cot⁻¹x

Important Identities

• sin⁻¹x + cos⁻¹x = π/2, −1 ≤ x ≤ 1
• tan⁻¹x + cot⁻¹x = π/2, x ∈ ℝ
• sec⁻¹x + cosec⁻¹x = π/2, |x| ≥ 1

Relations Between Inverse Functions

• sin⁻¹x = cosec⁻¹(1/x), |x| ≤ 1, x ≠ 0
• cos⁻¹x = sec⁻¹(1/x), |x| ≤ 1, x ≠ 0
• tan⁻¹x = cot⁻¹(1/x), x > 0
• If x < 0: tan⁻¹x = π + cot⁻¹(1/x)

Sum and Difference Formulae

Sine Inverse

• sin⁻¹x + sin⁻¹y = sin⁻¹( x√(1−y²) + y√(1−x²) )
  Condition: x² + y² ≤ 1
• sin⁻¹x − sin⁻¹y = sin⁻¹( x√(1−y²) − y√(1−x²) )

Cosine Inverse

• cos⁻¹x + cos⁻¹y = cos⁻¹( xy − √(1−x²)√(1−y²) )
• cos⁻¹x − cos⁻¹y = cos⁻¹( xy + √(1−x²)√(1−y²) )

Tangent Inverse

• tan⁻¹x + tan⁻¹y = tan⁻¹( (x + y)/(1 − xy) ) , xy < 1
• tan⁻¹x − tan⁻¹y = tan⁻¹( (x − y)/(1 + xy) )

Multiple Angle Results

• 2 sin⁻¹x = sin⁻¹(2x√(1−x²))
• 2 cos⁻¹x = cos⁻¹(2x² − 1)
• 2 tan⁻¹x = tan⁻¹(2x / (1 − x²))
• 3 sin⁻¹x = sin⁻¹(3x − 4x³)
• 3 cos⁻¹x = cos⁻¹(4x³ − 3x)
• 3 tan⁻¹x = tan⁻¹( (3x − x³)/(1 − 3x²) )

Useful Transformations

• tan⁻¹(x/√(a² − x²)) = sin⁻¹(x/a)
• tan⁻¹(√(1+x²) − √(1−x²) / √(1+x²) + √(1−x²)) = π/4 + (1/2)cos⁻¹x²

JEE Main Focus Tips

• Always check the principal value range
• Reduce expressions to standard inverse forms
• Be careful with sign and quadrant restrictions
• Use identities to simplify multi-term inverse expressions