6. Zadatak

Uočavamo da su svi faktori istog oblika pa najprije računamo

$$\begin{aligned} 1-\frac{\cos(60^\circ+k^\circ)}{\cos k^\circ} &=\frac{\cos k^\circ-\cos(60^\circ+k^\circ)}{\cos k^\circ}\\ &=\frac{2\sin30^\circ\sin(30^\circ+k^\circ)}{\cos k^\circ}\\ &=\frac{\sin(30^\circ+k^\circ)}{\cos k^\circ}\\ &=\frac{\cos(90^\circ-30^\circ-k^\circ)}{\cos k^\circ}\\ &=\frac{\cos(60^\circ-k^\circ)}{\cos k^\circ}. \end{aligned}$$

Dakle,

$$\begin{aligned} \prod_{k=1}^{59}\left(1-\frac{\cos(60^\circ+k^\circ)}{\cos k^\circ}\right) &=\frac{\cos59^\circ}{\cos1^\circ}\cdot\frac{\cos58^\circ}{\cos2^\circ}\cdots\frac{\cos1^\circ}{\cos59^\circ}\\ &=1. \end{aligned}$$