Trig question
So recently I have had an interest in some trig identities that aren't the normal ones you see on a unit circle that you learn from back in high school. And I've came across some interesting identities that involve angles that aren't for 15°, 18°, 30°, 45°, 60° and 90° and their sum/different/half angle angles . More specifically some interesting ones where it uses more abstract identities such as sin(x)+sin(60°-x)+sin(60°+x) = (sin[3x])/4 & other interesting properties. Ones that stood out to me though were ones that used stuff involving angles such as (2pi)/7 and it kind of made me wonder about how if those are able to have a radical form for it's cosine & sine, then how would one construct those mechanically.
To explain further
cos(pi/7)×cos([2pi]/7)×cos([3pi]/7)=1/8
and cos([2pi]/7) & cos([3pi]/7) can be rewritten in terms of cos(pi/7) to make it moreso an algebric solution to look for when subbing in a variable for cos(pi/7)
So how could one work out how to construct cos(pi/7) for it's respective radical form?