How to find the roots of $\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$? Find all $x$ in the interval $(0,\pi/2)$ such that $\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$.
The options are (i)$\pi/9,2\pi/7$, (ii)$\pi/36,11\pi/12$ (iii)$\pi/12,11\pi/36$ (iv) All
I have been able to find one value of $x$, $\pi/12$.  How do I find the other root(s)? 
My attempt:
$\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$
or, $\frac{\sin\pi/3-\sin\pi/6}{\sin x}+\frac{\cos\pi/6+\cos\pi/3}{\cos x}=2\sqrt{2}$
or, $\frac{\sin(\pi/4)cos(\pi/12)}{\sin x}+\frac{\cos(\pi/4)cos(\pi/12)}{\cos x}=\sqrt{2}$
or, $\sin(x+\pi/12)=\sin2x$
or, $x=\pi/12$

Be careful that your final equation has more potential solutions. The equation
$$ \sin \left(x + \frac{\pi}{12}\right) = \sin 2x$$
implies in fact
$$ x + \frac{\pi}{12} = 2x + 2k \pi$$
or
$$ x + \frac{\pi}{12} = \pi -  2x + 2k \pi.$$

Also recall that you can always check the number of solutions by intersecting
$$ \frac{\sqrt 3 -1}{Y} + \frac{\sqrt 3 +1}{X}=4 \sqrt 2$$
with the unit circle
$$X^2+Y^2 = 1.$$

