Expand the left hand side of the equation by using the binomial theorem, then group the real numbers together, and the imaginary numbers together.
The real part will be equal to , which is what you want.
(ii) If z = costheta + i sintheta, nÎZ^+, proof that z^n + z^-n = 2 cos ntheta, hence, Cos 4theta = 8 cos^4theta - 4cos2theta -3
and (because the cosine is even and the sine is odd)
add the two and you'll get what you want
as for the second part, just use let n=4 in the formula above. and use a similar method to 1)
(i'm looking for a simpler way...)
find the modulus of the RHS :(iii) Find the roots of z^3 = -1 + √3 i
find the value of theta