Canonical to Parametric, Ellipse Equation I've done some algebra tricks in this derivation and I'm not sure if it's okay to do those things.
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + \sin^2\theta$$
Can I really do this next step? 
$$\frac{x^2}{a^2} = \cos^2\theta\quad\text{and}\quad\frac{y^2}{b^2} = \sin^2\theta$$
$$x^2 = a^2\cos^2\theta\quad\text{and}\quad y^2 = b^2\sin^2\theta$$
Ignoring the negative numbers:
$$x = a\cos\theta\quad\text{and}\quad y = b\sin\theta$$

If you start with the line you wrote last and work your way backwards to the equation of the ellipse that you wrote at the top, then your logic is fine.
(But if your question gets edited after this answer appears, then this answer might be wrong after that.  It's happened here before.)
