I have the parameter form:
\begin{equation} x=\frac{1}{\cos t}\\ y=2\tan t \end{equation}
which I want to convert to implicit form. This turns out to be quite hard. One attempt is to set the square of each side of both eqns. and get:
\begin{equation} x^2=\frac{1}{\cos^2 t}\\ y^2=4\tan^2t \end{equation}
which is:
\begin{equation} x^2=\frac{1}{\frac{1}{2}(1+\cos2t)}\\ y^2=4\frac{1-\cos2t}{1+\cos2t} \end{equation}
For simplicity set $\cos2t=u$ and rearrange.
\begin{equation} x^2(1+u)=2\\ y^2(1+u)=4(1-u) \end{equation}
From here we get to
\begin{equation} (1+u)=\frac{2}{x^2}\\ (1+u)=\frac{4(1-u)}{y^2} \end{equation}
But this seems to go nowhere. Any hints?
$x =\sec t$ and $\sec^{2}t=1+\tan^{2}t$ so we get $x^{2}=1+\frac {y^{2}}4$. This is a hyperbola.