Question

Why does sin(arcsin(x))=x? And can you write arcsin(x) as 1/(sin x)?

Answer 1

arcsin also written as inverse sin ( sin^(-1) ) does not mean the same as the reciprocal of the sine function.

If you are given an angle (let's say in degrees) you can apply the sin function to it and get a number.

e.g. sin 30 = 0.5

Now if you take arcsin 0.5, you get 30 degrees. Arcsin is the inverse function of sine.

Answer 2

I know the notation \[\sin^{-1}(x)\]
is misleading...
But yeah this does not mean 1/sin(x) and that -1 is not an exponent in this case.
This is why some people prefer the notation arcsin(x)

Answer 3

But it is best to get use to both notations because they are both used