Question

Show cos 20 * cos 40 * cos 80 =1/8

Answer 1

supposed the value of (cos 20 * cos 40 * cos 80 = x)
multiply by sin20 to both sides, the equation above becomes
sin20 * cos 20 * cos 40 * cos 80 = x sin20

look at that sin20cos20 = 1/2 sin(2(20)) = 1/2 sin40, now we have
(sin20 * cos 20) * cos 40 * cos 80 = x sin20
1/2 sin40 * cos 40 * cos 80 = x sin20

and again by using double angle formula for sin, we can change sin40 * cos40 =
1/2 sin(2(40)) = 1/2 sin80, now we have
1/2 (sin40 * cos 40) * cos 80 = x sin20
1/2 * (1/2 sin80) * cos80 = xsin20
1/4 sin80 * cos80 = x sin20

and again again we can rwrite sin80 * cos80 becomes 1/2 sin(2(80)) = 1/2sin160
now we have :
1/4 (sin80 * cos80) = x sin20
1/4 * 1/2sin160 = x sin20

remember the identity :
sin(180-x) = sinx
therefore, sin160 = sin(180-20) = sin20

finally we have an equation :
1/4 * (1/2sin160) = x sin20
1/4 * 1/2 * sin20 = x sin20
1/8 sin20 = x sin20
divide by sin20 to both sides, finally we get

1/8 sin20/sin20 = x sin20/sin20
1/8 = x or
1/8 = cos 20 * cos 40 * cos 80

QED ^_^