Properties of inverse trigonometric functions II

Property VIII:

Property IX:

Property X:

Property XI:

Property XII:

Property XIII:

(i)                 sin^-¹ x = cos^-¹ √(1 - x²) = tan^-¹ [x / √(1 - x²)]
                       = cot^-¹ √(1 - x²) / x = sec^-¹ [1 / √(1 - x²)] 
                       = cosec^-¹ (1/x)

(ii)               cos^-¹ x = sin^-¹ √(1 - x²) = tan^-¹ [√(1 - x²) / x] 
                        = cot^-¹ [x / √(1 - x²)] = sec^-¹ 1/x
                        = cosec^-¹ [1/ √(1 - x²)]

(iii)             tan^-¹ x = sin^-¹ [x / √(1 + x²)] = cos^-¹ [x / √(1 + x²)]
                       = cot^-¹ (1/x) = sec^-¹ √(1 + x²)
                       = cosec^-¹ [√(1 + x²) / x]

Property XIV:

If x₁, x₂, … x_n ϵ R, then

tan^-¹ x₁ + tan^-¹ x₂ + … + tan^-¹ x_n = tan^-¹ [(s₁ - s₃ + s₅ - s₇ + ...) / (1 - s₂ + s₄ - s₆ + …)]

Where S_k = sum of the products of x₁, x₂, … x_n taken k at a time.