<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] We have $\dfrac{1-\cos^2\theta}{\cos\theta}=n\implies\sin^2\theta=n\cos\theta\  \  \  \ (1)$
$\iff\cos^2\theta+n\cos\theta-1=0\  \ \  \  (2)$
and multiply $\sin\theta$ with the both sides of $\sin \theta +\csc\theta = m$, $$\sin^2\theta=m\sin\theta-1\  \ \  \  (3)$$
$(1),(3)\implies n\cos\theta+1=m\sin\theta$
Squaring & on rearrangement, $(m^2+n^2)\cos^2\theta+2n\cos\theta+1-m^2=0\  \ \ \ (4)$
Solve $(2),(4)$ for $\cos^2\theta,\cos\theta$ and utilize the fact $\cos^2\theta=(\cos\theta)^2$
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