## Task $4 / 85$

Determine the value of the expression

$$
\cos 20^{\circ} \cdot \cos 40^{\circ} \cdot \cos 60^{\circ} \cdot \cos 80^{\circ}
$$

without using tables, calculators, or similar aids!
Let

$$
\cos 20^{\circ} \cdot \cos 40^{\circ} \cdot \cos 60^{\circ} \cdot \cos 80^{\circ}=x \quad ; \quad \sin 20^{\circ} \cdot \sin 40^{\circ} \cdot \sin 60^{\circ} \cdot \sin 80^{\circ}=y
$$

By multiplying, it follows that

$$
2 \sin 20^{\circ} \cos 20^{\circ} \cdot 2 \sin 40^{\circ} \cos 40^{\circ} \cdot 2 \sin 60^{\circ} \cos 60^{\circ} \cdot 2 \sin 80^{\circ} \cos 80^{\circ}=16 x y
$$

and because $2 \sin a \cos a=\sin (2 a), \sin 180^{\circ}-a=\sin a$ and $y \neq 0$

$$
\sin 40^{\circ} \cdot \sin 80^{\circ} \cdot \sin 60^{\circ} \cdot \sin 20^{\circ}=16 x y=y \rightarrow 16 x=1
$$

Thus,

$$
x=\cos 20^{\circ} \cdot \cos 40^{\circ} \cdot \cos 60^{\circ} \cdot \cos 80^{\circ}=0.0625
$$