Question

How do you prove `cos (pi/7) cos ((2pi)/7) cos ((3pi)/7) = 1/8`?

Answer 1

Since this is a proof, we do not have a short answer. See the explanation below.

Explanation:

First you should prove the following:

`\cos(2\pi/7)+\cos(4\pi/7)+\cos(6\pi/7)=-1/2`

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:

`1+\cos(2\pi/7)+\cos(4\pi/7)+\cos(6\pi/7)+\cos(8\pi/7)+\cos(10\pi/7)+\cos(12\pi/7)=0`

Since `\cos(2\pi-x)=\cos(x)` the sum above becomes:

`1+2\cos(2\pi/7)+2\cos(4\pi/7)+2\cos(6\pi/7)=0`

From that we then have the required result

`\cos(2\pi/7)+\cos(4\pi/7)+\cos(6\pi/7)=-1/2`

With that result proven, we are now ready to tackle the given product. First use the sum-product relation for cosines

`\cos(a)\cos(b)=(1/2)(\cos(a+b)+\cos(a-b))`

to get

`\cos(\pi/7)\cos(2\pi/7)=(1/2)(\cos(3\pi/7)+\cos(-\pi/7))`

Then:

`\cos(\pi/7)\cos(2\pi/7)\cos(3\pi/7) =(1/2)(\cos(3\pi/7)(\cos(3\pi/7)+(1/2)(\cos(-\pi/7)(\cos(3\pi/7))`

Apply the sum-product relation once more to each of the above terms to get

`(1/4)(\cos(6\pi/7)+\cos(0)+\cos(2\pi/7)+\cos((-4)\pi/7))`

Since

`\cos(0)=1` and `\cos(-4\pi/7)=\cos(4\pi/7)`
this can be rearranged to

`(1/4)(1+\cos(2\pi/7)+\cos(4\pi/7)+\cos(6\pi/7))=(1/4)(1-1/2)=1/8`

Answer 2

we shall use the formula

`2sintheta costheta=sin2theta`

`LHS=cos(pi/7)cos((2pi)/7)cos((3pi)/7)`

`=4/(8sin(pi/7))xx2sin(pi/7)cos(pi/7)cos((2pi)/7)cos((3pi)/7)`

`=2/(8sin(pi/7))xx2sin((2pi)/7)cos((2pi)/7)cos((3pi)/7)`

`=1/(8sin(pi/7))xx2sin((4pi)/7)cos((3pi)/7)`

`=1/(8sin(pi/7))xx2sin(pi-(3pi)/7)cos((3pi)/7)`

`=1/(8sin(pi/7))xx2sin((3pi)/7)cos((3pi)/7)`

`=1/(8sin(pi/7))xxsin((6pi)/7)`

`=1/(8sin(pi/7))xxsin(pi-(6pi)/7)`

`=1/(8sin(pi/7))xxsin(pi/7)`

`=1/8`

proved