Complex numbers and sequences

Question

Here is a question I came across in my Math textbook -

I have spent almost an hour on this and still can't work it out.

Any help will be highly appreciated.

Answer 1

Expanding $(1+z)^n$, we get $$ 1 + \binom n1z + \binom n2z^2 + \cdots + \binom nnz^n $$ Note that the sum we're after is exactly the imaginary part of this, by de Moivre's formula.

Let's raise the left-hand side of [3] to the power of $n$ as well, and see what we get, again using de Moivre's formula: $$ 2^n\cos^n\frac\theta2\left(\cos\frac{n\theta}2 + i\sin\frac{n\theta}2\right) $$ Now you just compare imaginary parts, and you're done.