The question is to find $x\in\left(0,\frac{\pi}{2}\right)$:
$$\frac {\sqrt3 – 1}{\sin x} + \frac {\sqrt3 + 1}{\cos x} = 4\sqrt2 $$
What I did was to take the $\cos x$ fraction to the right and try to simplify ;
But it looked very messy and trying to write $\sin x$ in terms of $\cos x$ didn't help.
Is there a more simple (elegant) way to do this.
Best Answer
$\sin(\frac{\pi}{12}) = \frac{\sqrt{3}-1}{2\sqrt{2}}$ and $\cos(\frac{\pi}{12}) = \frac{\sqrt{3}+1}{2\sqrt{2}}$. Plugging them into your equation yields $\sin(x+\frac{\pi}{12}) = \sin(2x)$. So $x = \frac{\pi}{12}$