If $\sin \theta +\cos \theta =p$ and $\sec \theta +\mathrm{cosec}\theta =q$, which relation between $p$ and $q$ is always true?

Squaring the first equation gives $p^2=1+2\sin \theta \cos \theta$, so $\sin \theta \cos \theta =\frac{p^2-1}{2}$. Also $q=\frac{1}{\cos \theta }+\frac{1}{\sin \theta }=\frac{\sin \theta +\cos \theta }{\sin \theta \cos \theta }=\frac{p}{\sin \theta \cos \theta }$. Substituting, $q=\frac{2p}{p^2-1}$, i.e. $q(p^2-1)=2p$. A sign slip on the cross term (writing $p^2=1-2\sin \theta \cos \theta$) is what produces the $p^2+1$ version.