Problem

The key is to rewrite the reciprocals of sines as a single equation and use the product-to-sum identity strategically. Notice that the right side has a common denominator structure—combine those fractions first, then use the sine addition formula or the identity $\sin A - \sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}$ to simplify. You're looking for an $n$ where the angles $\frac{\pi}{n}$, $\frac{2\pi}{n}$, and $\frac{3\pi}{n}$ satisfy a special trigonometric relationship, likely one that emerges from symmetry or a known identity.

The key is to rewrite the reciprocals of sines as a single equation and use the product-to-sum identity strategically. Notice that the right side has a common denominator structure—combine those fractions first, then use the sine addition formula or the identity $\sin A - \sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}$ to simplify. You're looking for an $n$ where the angles $\frac{\pi}{n}$, $\frac{2\pi}{n}$, and $\frac{3\pi}{n}$ satisfy a special trigonometric relationship, likely one that emerges from symmetry or a known identity.