I was trying to plot the parametric curve
$(cos(t^{2}), sin(t^{2}), 0 )$ with some online graphing tool.
Shouldn't the trace of this curve be the same as a circle, with only difference being how fast the curves "runs" through the circle with respect to t?
Yet the trace of the function I get with these online tools have points all over the interior of the trace of a circle.
Is my assumption incorrect?
You are right, the curve should be on the $z=0$ axis.
I have attached the output from desmos, an online tool to plot graph.
It holds due to the identity of $\sin^2(t)+\cos^2(t)=1$, and hence $\sin^2(t^2)+\cos^2(t^2)=1$ holds as well.