Cosine Chain

Let $\displaystyle P = \sin\left(\frac{2\pi}{15}\right)\sin\left(\frac{4\pi}{15}\right)\sin\left(\frac{8\pi}{15}\right)\sin\left(\frac{14\pi}{15}\right)$ and $\displaystyle Q = \cos\left(\frac{2\pi}{15}\right)\cos\left(\frac{4\pi}{15}\right)\cos\left(\frac{8\pi}{15}\right)\cos\left(\frac{14\pi}{15}\right)$ . Then $\displaystyle 16PQ=\sin\left(\frac{4\pi}{15}\right)\sin\left(\frac{8\pi}{15}\right)\sin\left(\frac{16\pi}{15}\right)\sin\left(\frac{28\pi}{15}\right)=P$ implies that $Q=\frac{1}{16}$ .

Hence, $A=1$ and $B=16$ which means that $B-A=15$ .