If ⁻¹(x - 2) + ⁻¹(x) + ⁻¹(x + 2) + ⁻¹(x + 4) = ⁻¹( k ) - 2 , then (2 , cosec ⁻¹ k - 1 ) =

Step-by-step solution

For the equation to be defined, the arguments of the inverse trigonometric functions must lie within their respective domains. For ⁻¹(x - 2) , we require -1 x - 2 1 1 x 3 . For ⁻¹(x) , we require -1 x 1 . The intersection of these two domains is x = 1 . Substituting x = 1 into the given equation: ⁻¹(1 - 2) + ⁻¹(1) + ⁻¹(1 + 2) + ⁻¹(1 + 4) = ⁻¹( k ) - 2 ⁻¹(-1) + ⁻¹(1) + ⁻¹(3) + ⁻¹(5) = ⁻¹( k ) - 2 - 2 + 0 + ⁻¹(3) + ⁻¹ ( 1 5 ) = ⁻¹( k ) - 2 ⁻¹(3) + ⁻¹ ( 1 5 ) = ⁻¹( k ) Using the formula ⁻¹x + ⁻¹y = ⁻¹ ( x + y 1 - xy )