Hello,

We know that , by de Moivre's theorem.

Expand the left hand side of the equation by using the binomial theorem, then group thereal numberstogether, and theimaginary numberstogether.

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

so

find the value of theta