If sin $\theta$ +cos $\theta$ = m and sec $\theta$ + cosec $\theta$ = n then 

$\begin{array}{l}Given\sin \theta +\cos \theta =m;\sec \theta +\cos ec\theta =n\\ \Rightarrow \frac{\sin \theta +\cos \theta }{\sin \theta \cos \theta }=n\Rightarrow \frac{m}{\sin \theta \cos \theta }=n\Rightarrow \sin \theta \cos \theta =\frac{m}{n}\\ we have{(\sin \theta +\cos \theta )}^2=1+2\sin \theta \cos \theta \Rightarrow m^2=1+2\frac{m}{n}\Rightarrow n(m^2-1)=2m.\end{array}$