When you equate the coefficients of z, you get the same result as (a)(iii) that is
-2\cos\frac{2\pi}{5}-2\cos\frac{4\pi}{5}=1
Using double angles and supplementary angles this is equivalent to
-2\left(2\cos^2\frac{\pi}{5}-1\right)+2\cos\frac{\pi}{5}=1
and the result follows...