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..

We have
$$\frac{1}{\cos (\theta)} - \cos (\theta) = n \qquad \qquad \qquad \qquad \sin (\theta) + \frac{1}{\sin(\theta)} = m$$
Let $x := \cos (\theta)$ and $y := \sin(\theta)$. Hence, $x^2 + y^2 = 1$. The two equations above can be rewritten as
$$\frac{1}{x} - x = n \qquad \qquad \qquad \qquad y + \frac{1}{y} = m$$
or, as follows
$$x^2 = 1 - n x \qquad \qquad \qquad \qquad y^2 = m y - 1$$
Since $x^2 + y^2 = 1$, we obtain the equation $m y - nx = 1$. Thus, we have the intersection of the unit circle and a line
$$x^2 + y^2 = 1\qquad \qquad \qquad \qquad m y - nx = 1$$
If $m, n$ are such that the intersection is not empty, then from $x, y$ we obtain the angle $\theta$. Note that the cardinality of the intersection is $0$, $1$, or $2$.
