Question

Let $E=\left(1-\frac{\cos 61^{\circ}}{\cos 1^{\circ}}\right)\left(1-\frac{\cos 62^{\circ}}{\cos 2^{\circ}}\right) \ldots\left(1-\frac{\cos 119^{\circ}}{\cos 59^{\circ}}\right)$ then $E$ is equal to

Answer 1

Given $I=\displaystyle\prod_{r=1}^{59}\left(1-\frac{\cos \left(60^{\circ}+r^{\circ}\right)}{\cos r^{\circ}}\right)$
$=\displaystyle\prod_{r=1}^{59} \frac{\sin \left(30^{\circ}+r^{\circ}\right)}{\cos r^{\circ}}$
$=\frac{\sin 31^{\circ} \cdot \sin 32^{\circ} \ldots \sin 89^{\circ}}{\cos 1^{\circ} \cdot \cos 2 \ldots \cos 59^{\circ}}=1$