Tenemos $\cos(\frac{\pi}{15})\cos(\frac{2\pi}{15})\ldots \cos(\frac{7\pi}{15})$ . Escribamos esto como $p=\cos(\frac{\pi}{15})\cos(\frac{2\pi}{15})\ldots \cos(\frac{7\pi}{15})$
Tenemos que multiplicar ambos lados de la ecuación por $$q= \sin(\frac{\pi}{15})\sin(\frac{2\pi}{15})\ldots \sin(\frac{7\pi}{15})$$
Ahora, $$p.q = \sin(\frac{\pi}{15})\sin(\frac{2\pi}{15})\ldots \sin(\frac{7\pi}{15})\cos(\frac{\pi}{15})\cos(\frac{2\pi}{15})\ldots \cos(\frac{7\pi}{15})$$
Multiplica ambos lados por $2^7$ $$2^7 p.q =[2 \sin(\frac{\pi}{15})\cos(\frac{\pi}{15})][2\sin(\frac{2\pi}{15})\cos(\frac{2\pi}{15})]\ldots [2\sin(\frac{7\pi}{15})\cos(\frac{7\pi}{15})]$$
$$2^7 p.q=\sin(\frac{2\pi}{15})\sin(\frac{4\pi}{15})\ldots \sin(\frac{14\pi}{15})$$
Ahora tenemos que aplicar la identidad $\sin\theta=\sin(\pi-\theta)$
$$2^7 p.q=\sin(\frac{\pi}{15})\sin(\frac{2\pi}{15})\ldots \sin(\frac{7\pi}{15})$$
Lo que significa $$2^7 p.q=q$$
Por lo tanto, $$p=\cos(\frac{\pi}{15})\cos(\frac{2\pi}{15})\ldots \cos(\frac{7\pi}{15})=\frac1{2^7}$$