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).