How to find the roots of $\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$?

Question

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, $x=\pi/12$

Answer 1

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 $$X^2+Y^2 = 1.$$

Answer 2

Use $$\sin15^{\circ}=\frac{\sqrt3-1}{2\sqrt2}$$ and $$\cos15^{\circ}=\frac{\sqrt3+1}{2\sqrt2}.$$ We obtain: $$\sin(15^{\circ}+x)=\sin2x.$$

Thus, $$15^{\circ}+x=2x+360^{\circ}k,$$ where $k$ is an integer number, or $$15^{\circ}+x=180^{\circ}-2x+360^{\circ}k.$$ Can you end it now?