IMO 1967 LL HUN21

Origin: HUN

Problem

Without using any tables, find the exact value of the product P = cos \pi 15 cos 2\pi 15 cos 3\pi 15 cos 4\pi 15 cos 5\pi 15 cos 6\pi 15 cos 7\pi 15 .

Solution

Using the formula cos x cos 2x cos4x \cdot \cdot \cdot cos 2n−1x = sin 2nx 2n sin x, which is shown by simple induction, we obtain cos \pi 15 cos 2\pi 15 cos 4\pi 15 cos 7\pi 15 = −cos \pi 15 cos 2\pi 15 cos 4\pi 15 cos 8\pi 15 = 1 16, cos 3\pi 15 cos 6\pi 15 = 1 4, cos 5\pi 15 = 1 2. Multiplying these equalities, we get that the required product P equals 1/128.