Inverse sin(x) should be cosec(x)?

Why is sin^-1(x) equal to arcsin(x), when the power of -1 should flip the expression to give 1/sin(x) which is cosec(x)?

I think you maybe confusing a number of things here. Sin^-1(x) and Arcsin(x) and Inverse sin(x)=THE SAME THING!!! Inverse sin(x) is often written as: Sin^-1. The power sign "^" in NOT meant to raise sin(x) to -1!!!!!!. Inverse sine or Arcsine is written like that for convenience ONLY.

In this case the -1 does not mean flip (find reciprocal) 

It mean find the angle whose sin is x ..     which is also asin x

\(If \qquad sin^{-1}(x)=\theta \qquad then \qquad sin \theta =x\\ and \qquad sin^{-1}(x)=asin\;(x)\)

I think the asin notation is much better because the -1 notation is very confusing.

If you want the reciprocation you need to write

\((sin\theta)^{-1} = \frac{1}{sin(\theta)}=Cosec(\theta)\)

Thanks, I knew that sin^-1(x) = arcsin(x), but I didnt realise the -1 was not a power but rather a visual representation, thanks for clearing that up.