Simplifying $\frac{p^2q^2(1-\epsilon^2\cos^2t)}{p^2\cos^2t+q^2\sin^2t}$, where $\epsilon=\sqrt{1-(q/p)^2}$ I am currently trying to show that the product of the distances from the focis of an ellipse to the tangent line at any point of the ellipse is a constant. While I thought that the computation is a straightforward plug-in and be done with it, this turned out to be harder than expected.
I know that the end result should be $q^2$, but I have no idea on how to manipulate the expression
$$\frac{p^2q^2(1 - \epsilon^2\cos^2(t))}{p^2\sin^2(t) + q^2\cos^2(t)}$$ where $p > q > 0, \epsilon = \sqrt{1 - (q/p)^2}$.
Edit: I had mixed up $\sin$ and $\cos$ in the denominator, so instead of $p^2\cos^2(t) + q^2\sin^2(t)$ the expression should be $p^2\sin^2(t) + q^2\cos^2(t)$.

Marcos already calculated value of your expression if $t=0$ as $q^4/p^2$. If $t=\pi/2$ expression is equal to $p^2$. If $p \ne q$, these values are different, that is in this case your expression is not a constant. If $p=q$ the expression is equal to $p$ or $q$ for any $t$.
