Eliminate $\theta$ 
Eliminate $\theta$ in
$$\sin \theta + \mbox{cosec} \, \theta = m$$
$$\sec \theta - \cos \theta = n$$

My approach-
I multiplied the first equation by $\sin \theta$ and the second equation by $\cos \theta$ but it doesn't give me the desired answer..

Squaring and adding,we get
$1+cosec^2\theta+\sec^2\theta=m^2+n^2.....(1)$
We need to find $cosec\theta$ and $\sec\theta$ from the given equations
First equation becomes $\frac{1}{\csc\theta}+\csc\theta=m$
$cosec^2\theta-m$ $cosec\theta+1=0$
Similarly second equation becomes
$\sec^2\theta-n\sec\theta-1=0$
Solving these using quadratic formula,
$cosec\theta=\frac{m\pm\sqrt{m^2-4}}{2}$
$\sec\theta=\frac{n\pm\sqrt{n^2+4}}{2}$
Put these values in equation $(1)$.
