Question

Is arcsin(x) = csc(x) true?

Answer 1

No. This is confusing `sin^-1(x)` with `(sin(x))^-1`.

Explanation:

`arcsin(x) = sin^-1(x)` is the inverse function of the function `sin(x)`

That is:

If `x in (-pi/2, pi/2)`, then `arcsin(sin(x)) = x`

If `x in [-1, 1]` then `sin(arcsin(x)) = x`

On the other hand:

`csc(x) = (sin(x))^(-1) = 1/sin(x)` is the reciprocal of the `sin` function.

I think some of the blame for this confusion has to lie with the common convention of writing `sin^2(x)` to mean `sin(x)^2`. So when you have `csc(x) = 1/sin(x) = sin(x)^(-1)` you might think that we would also write that as `sin^(-1)(x)`, but that's reserved for `arcsin(x)`.