Explain the difference between 1/(sinx) and ((sin^-1)x)

1/sin(x) is the inverse of the entire value of sin(x), while sin⁻¹(x) is the inverse of the value of x. These are different operations so yield different answers. For example, let x = π/2
1/sin(π/2) = 1
sin⁻¹(π/2) does not exist because π/2 lies outside the range of sin(x).