TOPICS
Formulae
Principal Values :-
1. sin-1 x : Domain = [-1, 1] & Range = [-π/2, π/2]
2. cos-1 x : Domain = [-1, 1] & Range = [0, π]
3. tan-1 x : Domain = R & Range = (-π/2, π/2)
4. cot-1 x : Domain = R & Range = (0, π)
5. sec-1 x : Domain = R - (-1, 1) & Range = [0, π] - {π/2}
6. cosec-1 x : Domain = R - (-1, 1) & Range = [-π/2, π/2] - {0}
    
Properties :-
1. sin-1 (1/x) = cosec-1 x, x ≥ 1 or x ≤ -1
2. cos-1 (1/x) = sec-1 x, x ≥ 1 or x ≤ -1
3. tan-1 (1/x) = cot-1 x, x > 0
4. cot-1 (1/x) = tan-1 x, x > 0
5. sec-1 (1/x) = cos-1 x, x ≥ 1 or x ≤ -1
6. cosec-1 (1/x) = sin-1 x, x ≥ 1 or x ≤ -1
    
Properties :-
1. sin-1 (-x) = -sin-1 x, x ∈ [-1, 1]
2. cos-1 (-x) = π - cos-1 x, x ∈ [-1, 1]
3. tan-1 (-x) = -tan-1 x, x ∈ R
4. cot-1 (-x) = π - cot-1 x, x ∈ R
5. sec-1 (-x) = π - sec-1 x, |x| ≥ 1
6. cosec-1 (-x) = -cosec-1 x, |x| ≥ 1
    
Properties :-
1. sin-1 x + cos-1 x = π/2, x ∈ [-1, 1]
2. cos-1 x = π/2 - sin-1 x, x ∈ [-1, 1]
3. sin-1 x = π/2 - cos-1 x, x ∈ [-1, 1]
4. tan-1 x + cot-1 x = π/2, x ∈ R
5. cot-1 x = π/2 - tan-1 x, x ∈ R
6. tan-1 x = π/2 - cot-1 x, x ∈ R
7. sec-1 x + cosec-1 x = π/2, |x| ≥ 1
8. cosec-1 x = π/2 - sec-1 x, |x| ≥ 1
9. sec-1 x = π/2 - cosec-1 x, |x| ≥ 1
    
Properties :-
1. tan-1 x + tan-1 y = tan-1[(x + y) ÷ (1 - xy)]
2. tan-1 x - tan-1 y = tan-1[(x - y) ÷ (1 + xy)]
3. sin-1 x + sin-1 y = sin-1(x√1 - y² + y√1 - x²)
4. sin-1 x - sin-1 y = sin-1(x√1 - y² - y√1 - x²)
5. cos-1 x + cos-1 y = cos-1(xy - √1 - y²1 - x²)
6. cos-1 x - cos-1 y = cos-1(xy + √1 - y²1 - x²)
    
Properties :-
1. sin-1 (sin x) = x
2. cos-1 (cos x) = x
3. tan-1 (tan x) = x
4. cot-1 (cot x) = x
5. sec-1 (sec x) = x
6. cosec-1 (cosec x) = x
    
Properties :-
1. sin-1 (sin x) = x or sin (sin-1 x) = x
2. cos-1 (cos x) = x or cos (cos-1 x) = x
3. tan-1 (tan x) = x or tan (tan-1 x) = x
4. cot-1 (cot x) = x or cot (cot-1 x) = x
5. sec-1 (sec x) = x or sec (sec-1 x) = x
6. cosec-1 (cosec x) = x or cosec (cosec-1 x) = x
    
Properties :-
1. 2tan-1 x = tan-1[2x ÷ (1 - x²)] = x
2. 2tan-1 x = sin-1(2x ÷ (1 + x²)] = x
3. 2tan-1 x = cos-1[(1 - x²) ÷ (1 + x²)] = x
4. 2sin-1 x = sin-1[2x √1 - x²]
5. 2cos-1 x = cos-1(2x² - 1) or sin-1[2x √1 - x²]
6. 3tan-1 x = tan-1[(3x - x³) ÷ (1 - 3x²)]
7. 3sin-1 x = sin-1[3x - 4x³]
8. 3cos-1 x = cos-1[4x³ - 3x]
    
Properties :-
1. sin-1 x = cos-1(√1 - x²)
          or tan-1(x ÷ √1 - x²)
          or cot-1 (√1 - x² ÷ x)
          or sec-1(1/√1 - x²)
          or cosec-1(1/x)

2. cos-1 x = sin-1(√1 - x²)
          or cot-1(x ÷ √1 - x²)
          or tan-1 (√1 - x² ÷ x)
          or cosec-1(1/√1 - x²)
          or sec-1(1/x)

3. tan-1 x = sec-1(√1 + x²)
           or sin-1(x ÷ √1 + x²)
           or cosec-1 (√1 + x² ÷ x)
           or cos-1(1/√1 + x²)
           or cot-1(1/x)
    
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