# Question #67c01

Jun 4, 2017

See below.

#### Explanation:

Set

$\cos 20 \cos 40 \cos 60 \cos 80 = a$

multiply both sides by $\sin 20$.

$\sin 20 \cos 20 \cos 40 \cos 60 \cos 80 = a \sin 20$

By double angle formula,

$\sin 20 \cos 20 = \frac{1}{2} \sin 40$

We can continue this pattern.

$\frac{1}{2} \sin 40 \cos 40 \cos 60 \cos 80 = a \sin 20$

$\frac{1}{4} \sin 80 \cos 80 \cos 60 = a \sin 20$

$\frac{1}{8} \sin 160 \cos 60 = a \sin 20$

$\cos 60$ is simply $\frac{1}{2}$

So,

$\frac{1}{16} \sin 160 = a \sin 20$

However,

$\sin 160 = \sin 20$

So, $a = \frac{1}{16}$, and we have proven the statement.