I'm trying to express cos(pi/7) with radicals - that is, basic operations (addition, division, etc) and square and cube roots (or even higher roots if necessary).

I got this far:

If and , then

Although this seems a complex expression, the imaginary part of x is actually zero, so x is in fact a real number (obviously, since cos(pi/7) is real). But I can't extract the real part as a "really real" radical expression (without i, that is).