Question #67c01

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Answer 1 · 2017-06-04T12:58:32

See below.

Set

$cos20cos40cos60cos80=a$

multiply both sides by $sin20$ .

$sin20cos20cos40cos60cos80=asin20$

By double angle formula,

$sin20cos20=1/2sin40$

We can continue this pattern.

$1/2sin40cos40cos60cos80=asin20$

$1/4sin80cos80cos60=asin20$

$1/8sin160cos60=asin20$

$cos60$ is simply $1/2$

So,

$1/16sin160=asin20$

However,

$sin160=sin20$

So, $a=1/16$ , and we have proven the statement.