Derive an ellipse tangent without losing its vertical cases

The curve $C$ has $x=2\cos t$, $y=\sin t$, $0\le t<2\pi$.

(i) Find its cartesian equation.2 marks

(ii) Sketch $C$.2 marks

(iii) Show that the tangent at the point with parameter $t$ is $(\cos t)x+(2\sin t)y=2$.3 marks

(iv) Hence find the tangents at $P$, where $t=7\pi/6$, and $Q$, where $t=11\pi/6$.2 marks