Overview of Inverse Trigonometric Functions

If

• sin θ = x ⟹ θ = sin⁻¹x...θ ∈ [−π/2, π/2]

• cos θ = x ⟹ θ = cos⁻¹x...θ ∈ [0, π]

• tan θ = x ⟹ θ = tan⁻¹x...θ ∈ (−π/2, π/2)

sin⁻¹x, cos⁻¹x, tan⁻¹x, etc. are called inverse trigonometric functions.

(A) Direct identities

• sin(sin⁻¹x) = x, |x| ≤ 1

• cos(cos⁻¹x) = x, |x| ≤ 1

• tan(tan⁻¹x) = x, x ∈ ℝ

• cot(cot⁻¹x) = x, x ∈ ℝ

• sec(sec⁻¹x) = x, |x| ≥ 1

• cosec(cosec⁻¹x) = x, |x| ≥ 1

(B) Inverse of trigonometric expressions

Valid ONLY in principal value range:

• sin⁻¹(sin θ) = θ, θ ∈ [−π/2, π/2]

• cos⁻¹(cos θ) = θ, θ ∈ [0, π]

• tan⁻¹(tan θ) = θ, θ ∈ (−π/2, π/2)

• cosec⁻¹ x = sin⁻¹1 (1/x), x ∈ R − (−1, 1)

• sec⁻¹ x = cos⁻¹ (1/x), x ∈ R − (−1, 1)

• cot⁻¹ x = tan⁻¹ (1/x), for x > 0

• cot⁻¹ x = π + tan⁻¹ (1/x), for x < 0
  [only if tan⁻¹ is taken in (−π/2, π/2)]

• sin⁻¹x = tan⁻¹( x / √(1−x²) ), |x| < 1

• cos⁻¹x = tan⁻¹( √(1−x²) / x ), x > 0

• tan⁻¹x = sin⁻¹( x / √(1+x²) ), ∀ x

• tan⁻¹x = cos⁻¹( 1 / √(1+x²) ), x ≥ 0

(A) tan⁻¹ formulas

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

• tan⁻¹x + tan⁻¹y = π + tan⁻¹( (x+y)/(1−xy) ), if x,y > 0 & xy > 1

• tan⁻¹x − tan⁻¹y = tan⁻¹( (x−y)/(1+xy) ) if x,y> -1

(B) sin⁻¹ formulas  

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

• sin⁻¹x − sin⁻¹y
  = sin⁻¹( x√(1−y²) − y√(1−x²) )

(C) cos⁻¹ formulas

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

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

• sin⁻¹(−x) = −sin⁻¹x, |x| ≤1

• tan⁻¹(−x) = −tan⁻¹x, x ∈ ℝ

• cosec⁻¹(−x) = −cosec⁻¹x, |x| ≥ 1

• cos⁻¹(−x) = π − cos⁻¹x, |x| ≤ 1

• sec⁻¹(−x) = π − sec⁻¹x, |x| ≥ 1

• cot⁻¹(−x) = π − cot⁻¹x, x ∈ ℝ

• sin⁻¹x + cos⁻¹x = π/2, |x| ≤1

• tan⁻¹x + cot⁻¹x = π/2, x ∈ ℝ

• sec⁻¹x + cosec⁻¹x = π/2, |x| ≥ 1

sin⁻¹ x = cos⁻¹ (√(1 − x²)), 0 ≤ x ≤ 1

cos⁻¹ x = sin⁻¹ (√(1 − x²)), 0 ≤ x ≤ 1

cos(sin⁻¹ x) = sin(cos⁻¹ x) = √(1 − x²), |x| ≤ 1

2 sin⁻¹ x = sin⁻¹ (2x√(1 − x²))

3 sin⁻¹ x = sin⁻¹ (3x − 4x³)

2 cos⁻¹ x = cos⁻¹ (2x² − 1)

3 cos⁻¹ x = cos⁻¹ (4x³  − 3x)

3 tan⁻¹ x = tan⁻¹ ((3x − x³ )/(1 − 3x²))