C = \{(x, y) \in \mathbb{R}^2 \mid y^2 = x^6(1 - x^2)\}.
(a) Is C a curve? If yes, why yes? If no, why not?
(b) Find a parametrisation of C and sketch C. Be sure to specify the domain of your
parametrisation.
Hint: a nice set of identities to consider are $\cos^2(\frac{t}{2}) = \frac{1 + \cos t}{2}$, $\sin^2(\frac{t}{2}) = \frac{1 - \cos t}{2}$, and
$\sin t = 2\sin(\frac{t}{2})\cos(\frac{t}{2})$.
(c) Find the tangent line to C at $(\frac{\sqrt{2}}{2}, \frac{1}{4})$.