Property II:

(i) sin (sin^-1 x) = x= cos (cos^-1 x),                for all x ∈ [-1, 1]

Property III:

(i) sin^-1 (-x) = -sin^-1 (x),                  for all x ∈ [-1, 1]

(ii) cos^-1 (-x) = π - cos^-1 (x),           for all x ∈ [-1, 1]

(iii) tan^-1 (-x) = -tan^-1 x                   for all x ∈ R

(iv) cosec^-1 (-x) = -cosec^-1 x,         for all x ∈ [-∞, -1] υ [1, ∞)

(v) sec^-1 (-x) = π - sec^-1 x,              for all x ∈ [-∞, -1] υ [1, ∞)

(vi) cot^-1 (-x) = π - cot^-1 x,              for all x ∈ R

Property IV:

(i) sin^-1 x + cos^-1 x = (π/2),            for all x ∈ [-1, 1]

(ii) tan^-1 x + cot^-1 x = (π/2),           for all x ∈ R

(iii) sec^-1 x + cosec^-1 x = (π/2),      for all x ∈ (-∞, -1] υ [1, ∞)

Property V:                     

(i) sin^-1 (1/x) = cosec^-1 x,               for all x ∈ (-∞, -1] υ [1, ∞)

(ii) cos^-1 (1/x) = sec^-1 x,                 for all x ∈ (-∞, -1] υ [1, ∞)

Property VI:

(i) sin^-1 x = cos^-1 (1 – x²)^1/2 = tan^-1 [x/(1 – x²)^1/2]

= cot^-1 [((1 – x²)^1/2)/x] = sec^-1 [1/(1 – x²)^1/2] = cosec^-1 (1/x), x ∈ (0, 1)

(ii) cos^-1 x = sin^-1 (1 – x²)^1/2 = tan^-1 [((1 – x²)^1/2)/x]

= cot^-1 [x/(1 – x²)^1/2] = sec^-1 (1/x) = cosec^-1 [1/(1 – x²)^1/2], x ∈ (0, 1)

(iii) tan^-1 x = sin^-1 [x/(1 + x²)^1/2] = cos^-1 [1/(1 + x²)^1/2]

= cot^-1 (1/x) = sec^-1 [(1 + x²)^1/2] = cosec^-1 [((1 + x²)^1/2)/x], x > 0

Property VII:

(i) sin (cos^-1 x) = cos (sin^-1 x) = (1 – x²)^1/2, -1 ≤ x ≤ 1.

(ii) tan (cot^-1 x) = cot (tan^-1 x) = (1/x), x ∈ R, x ≠ 0.

(iii) cosec (sec^-1 x) = sec (cosec^-1 x) =. |x| > 1.