Conversion of Parametric Equation to a Rectangular Equation I was looking for the rectangular equation of the given parametric equation $$ x = \tan \theta$$
$$ y = \tan 2\theta$$
My work:
$$x = \tan \theta$$
$$\theta = \arctan x$$
Then, substituting it to $ y = \tan 2\theta$, it becomes:
$$y = \tan 2\theta$$
$$y = \tan (2(\arctan x))$$
At this point, I'm stuck. How do you get the Rectangular equation of the above Parametric equation?

From trigonometry we know that $\displaystyle \tan(\alpha+\beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}.$
If $\alpha$ happens to be the same as $\beta$ then this says $\displaystyle\tan(\alpha+\alpha) = \frac{\tan\alpha + \tan\alpha}{1-\tan\alpha\tan\alpha} = \frac{2\tan\alpha}{1 - \tan^2\alpha}.$
In other words $\displaystyle\tan(2\alpha) = \frac{2\tan\alpha}{1-\tan^2\alpha}.$
That is the double-angle formula for the tangent function.
So $\displaystyle y = \tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta} = \frac{2x}{1-x^2}.$
So you're looking for the graph of $y = \dfrac{2x}{1-x^2}.$
(Note that $\tan\theta$ goes all the way from $-\infty$ to $+\infty,$ and therefore so does $x.$) 
