<s>[INST] 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..
 [/INST] using shorthand 
$$ s = \sin \theta; c = \cos \theta; s2 = \sin 2 \theta; c2 = \cos 2 \theta; $$
Square and add
$$ s^2 +1/s^2 + c^2 + 1/c^2 = m^2+n^2$$
or
$$ sc= \frac{1}{\sqrt{m^2n^2-1}}, \, s2= \frac{2}{\sqrt{m^2+n^2-1}}  $$
$$c2 = \sqrt{1-s2^2}=\sqrt{ \frac{ m^2+n^2-5}{m^2+n^2-1}}$$
$$s = \sqrt { \frac{ 1-c2}{2}}$$
Plug into first given relation
$$\sqrt { \frac{ 1-c2}{2}}  +  \sqrt { \frac{2}{ 1-c2}} = m $$
Simplify. There are two signs for every $\pm$, so there are eight eliminants to be considered.
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