eliminating t in a paramentric equation t (stewart pg 652, example #2)

What curve is represented by the parametric equations x=cost and y=sint

0<=t<=2pie ?

The solution says the following: "If we plot points, it appears that the curve is a circle. We can confirm this by eliminating t.

Observe that x^2 + y^2 = cos^2t + sin^2t = 1 "

But by my reasoning, if we eliminate t we would be left with y = sin (arccos x)

heres how i eliminated t:

x = cos t

t = arccos x

then after inserting into the y equation i'm left with: y = sin (arccos x)

I of course understand that we have equation of circle and the trig identity, but I dont understand how they're arriving at the identify by eliminating t??

please help

-tanks!