Integral solutions to inverse trigonometric equation. The original question is this : I have to find number of ordered pairs of integral solutions to this :
$$\tan^{-1}{x} + \cos^{-1}{\frac{y}{\sqrt{1+y^2}}} = \sin^{-1}{\frac{3}{\sqrt{10}}} $$
I rearranged this as $\tan^{-1}(\frac{1}{y}) = tan^{-1}{3} - \tan^{-1}x $
And then $\frac{1}{y}=\frac{3-x}{1+3x}$.
Finally reduced it to $$(1+3x).(1-3y)=-10xy$$
The problem is $-10xy = -1.2.5.x.y$
So, do I really have to check $32$ cases? I know some of them can be discarded easily, but still, checking all will take a lot of time.
My question is, is there a different method to solve this? Or is there an easier way to check the cases?
Note : The time constraint of the question is ideally 3-4 minutes. This is a question for IIT-JEE which is an entrance exam for engineering in India.

$$(1+3x)(1-3y)+10xy=0$$
is equivalent to:
$$ 1+3x-3y+xy = 0 $$
or to:
$$ (y+3)(3-x)=10=\pm 1\cdot \pm 10 = \pm 2\cdot\pm 5. $$
