z = 1 + root 3 i

z bar = 1 - root3 i

Converting this to cis form, z bar = 2 cis (-pi/3)

Using De Moivre's Theorem, (z bar)^5 = (2^5) cis 5*(-pi/3) = 32 cis (pi/3)

If w = root 2 cis (pi/4)

w/z in polar form would be (32/ root 2) cis (pi/3 - pi/4) = 16 root 2 cis (pi/12)

w/z in cartesian form would be:

w in cartesian form is root2 (cos pi/4 + i sin pi/4) = root 2 [(1/root 2) + i (1/root 2)] = 1 + i

w/z = (1+i)/(1 + root 3 i)

Realising the denominator would give you w/z = (1 + root 3)/4 + (1 - root 3)/4 * i

and thus deduce an exact value for cos(pi/12).

We know that 16 root 2 cis (pi/12) = 16 root 2 [cos (pi/12) + i sin (pi/12)] = (1 + root 3)/4 + (1 - root 3)/4 * i

So 16 root 2 * cos (pi/12) = (1 + root 3)/4

cos (pi/12) = (1 + root 3)/4 divided by 16 root 2...

Simplify and realise the answer.

Done!