How do you prove #cos (pi/7) cos ((2pi)/7) cos ((3pi)/7) = 1/8#?
Since this is a proof, we do not have a short answer. See the explanation below.
First you should prove the following:
To do that, draw a regular heptagon ABCDEFG with unit sides. Think of the sides as a vector, from A to B, then from B to C, etc. The vectors form a cycle, so from the "head to tail" rule the resultant is zero. At the same time we can define an "x-axis" from A to B and work out the components of each side's vector along that axis. Add them up and the sum must match the zero resultant:
From that we then have the required result
With that result proven, we are now ready to tackle the given product. First use the sum-product relation for cosines
Apply the sum-product relation once more to each of the above terms to get
this can be rearranged to
we shall use the formula