TITLE: $\sec\theta+\tan\theta=p$ and $\sec\theta\tan\theta=q$. Eliminate $\theta$ to form a equation between $p$ and $q$.
QUESTION [10 upvotes]: $\sec\theta+\tan\theta=p$ and $\sec\theta\tan\theta=q$. Eliminate $\theta$ to form a equation between $p$ and $q$.

$\sec\theta+\tan\theta=p$
$(\sec\theta+\tan\theta)^2=p^2$
$\sec^2\theta+\tan^2\theta+2\tan\theta\sec\theta=p^2$
$\sec^2\theta+\tan^2\theta+2q=p^2$
$1+2\tan^2\theta+2q=p^2$
I am stuck here. Please help me. Thanks.

REPLY [2 votes]: $$\sec\theta+\tan\theta=p\iff\sec\theta-\tan\theta=\dfrac1p$$
Now $(\sec\theta+\tan\theta)^2-(\sec\theta-\tan\theta)^2=4\sec\theta\tan\theta$
Replace the values of $\sec\theta+\tan\theta,\sec\theta-\tan\theta, \sec\theta\tan\theta$