Keep the trigonometric range and solve an intersection exactly

(a) Find the cartesian equation for $x=\cos\theta$, $y=2\sin^2\theta$.2 marks

(b) A curve has $x=t^3+t$, $y=2t-1$, $-1\le t\le1$.

(i) Sketch the curve, indicating its axial intercepts.3 marks

(ii) Without using a calculator, find its intersection with $y=\tfrac12$.2 marks