Let \(z=\cos \theta+\mathrm{i} \sin \theta\). Use the binomial expansion of \((1+z)^{n}\), where \(n\) is a positive integer, to show that
\(\binom{n}{1} \cos \theta+\binom{n}{2} \cos 2 \theta+\ldots+\binom{n}{n} \cos n \theta=2^{n} \cos ^{n}\left(\frac{1}{2} \theta\right) \cos \left(\frac{1}{2} n \theta\right)-1 .\)

Find
\(\binom{n}{1} \sin \theta+\binom{n}{2} \sin 2 \theta+\ldots+\binom{n}{n} \sin n \theta .\)