Finding Solutions to Trigonometric Equation Find all $x$ in the interval (0, $\frac{\pi}{2}$) such that $$\frac{\sqrt{3}-1}{\sin x} + \frac{\sqrt{3}+1}{\cos x} = 4\sqrt{2}.$$

Rewrite it in the form
$$2\sqrt2\left(\frac{\sqrt3-1}{2\sqrt2}\cos x+\frac{\sqrt3+1}{2\sqrt2}\sin x\right)=2\sqrt2\sin 2x.$$
For $\phi=\arcsin\frac{\sqrt3-1}{2\sqrt2}$ it implies
$$\sin(x+\phi)=\sin 2x,$$
i.e. $x+\phi=2x+2\pi n$ or $x+\phi=\pi-2x+2\pi n$, $n\in\Bbb Z$. Therefore, the only solutions in $(0,\pi/2)$ are $\phi$ and $\frac{\pi-\phi}{3}$.
