When to ignore the 0 values of a variable when dividing an equation by a variable

Question

So, I was watching this video of parametric equations and the elimination of the parameter. When you are solving a system of equations,you basically cannot divide by any of the variables, in case that variable is taking a value of 0, but in this video they gave this example: $x=\sin(t)$, $y=\sin(2t)$ for t in the interval $t\epsilon [0,2\pi]$ Using the sine of double angles identity, $y=2\sin(t)\cos(t)$, which implies $y=2x\cos(t)$, but here comes the problem: in order to use the identity $\sin^2(t)+\cos^2(t)=1$ they solve the equation like this: $\cos(t)=\frac{y}{2x}$ so now the parameter can be eliminated $$\sin^2(t)+\cos^2(t)=1=x^2+(\frac{y}{2x})^2$$ That finally results in the form $y^2+4x^4-4x^2=0$ but we totally ignored the values of x that make $\cos(t)$ undetermined. When can I actually do this?

Answer 1

The argument you are giving proves that the portion of the curve $(\sin t,\sin (2t))$, $t\in[0,2\pi]$ for which $\sin t\ne0$ parametrizes a portion of the locus $\{(x,y)\in\Bbb R^2\,:\, y^2+4x^4-4x^2=0\}$. In order to prove that the whole curve parametrizes a portion of said locus, you only have to check the equation for the points you were excluding when you divided by $x$ before.

Answer 2

Slope $\dfrac{dy}{dx}$ at $t=0$ can be either positive or negative, so it is a self intersection cross over point. The two values should be included for $\cos t $.