Calculation of sums and products with the help of complex numbers

$1-\binom {n}{2}+\binom {n}{4}-\binom {n}{6}+\ldots$;

$\binom {n}{1}-\binom {n}{3}+\binom {n}{5}-\binom {n}{7}+\ldots$;

$1+\binom {n}{4}+\binom {n}{8}+\ldots ;$

$\binom {n}{1}+\binom {n}{5}+\binom {n}{9}+\ldots$

$\cos x+\cos 2 x+\ldots +\cos n x=\frac{\sin \frac{n x}{2} \cos \frac{(n+1) x}{2}}{\sin \frac{x}{2}} \quad (x \neq 2 k \pi , k \in \mathbb {Z})$;

$\sin x+\sin 2 x+\ldots +\sin n x=\frac{\sin \frac{n x}{2} \sin \frac{(n+1) x}{2}}{\sin \frac{x}{2}} \quad (x \neq 2 k \pi , k \in \mathbb {Z})$;

$\cos \frac{\pi }{n}+\cos \frac{3 \pi }{n}+\cos \frac{5 \pi }{n}+\ldots +\cos \frac{(2 n-1) \pi }{n}=0 ;$

$\sin \frac{\pi }{n}+\sin \frac{3 \pi }{n}+\sin \frac{5 \pi }{n}+\ldots +\sin \frac{(2 n-1) \pi }{n}=0$;

$\frac{1}{n} \sum_{k=0}^{n-1}\left(x+\varepsilon_{k} y\right)^{n}=x^{n}+y^{n} \quad \left(\varepsilon_{0}, \varepsilon_{1}, \ldots , \varepsilon_{n-1}\right.$ are the roots of degree $n$ of unity);

$x^{2 n+1}-1=(x-1) \prod_{k=1}^{n}\left(x^{2}-2 x \cos \frac{\pi k}{2 n+1}+1\right) ;$

$x^{2 n}-1=\left(x^{2}-1\right) \prod_{k=1}^{n-1}\left(x^{2}-2 x \cos \frac{\pi k}{n}+1\right)$;

$\prod_{k=1}^{n-1} \sin \frac{\pi k}{2 n}=\frac{\sqrt{n}}{2^{n-1}}$;

$\prod_{k=1}^{n} \sin \frac{\pi k}{2 n+1}=\frac{\sqrt{2 n+1}}{2^{n}}$.

$1+\binom {n}{3}+\binom {n}{6}+\ldots =\frac{1}{3}\left(2^{n}+2 \cos \frac{\pi n}{3}\right)$;

$\binom {n}{1}+\binom {n}{4}+\binom {n}{7}+\ldots =\frac{1}{3}\left(2^{n}+2 \cos \frac{(n-2) \pi }{4}\right)$;

$\binom {n}{2}+\binom {n}{5}+\binom {n}{8}+\ldots =\frac{1}{3}\left(2^{n}+2 \cos \frac{(n-4) \pi }{3}\right)$;

$2 \cos m x=(2 \cos x)^{m}-\frac{m}{1}(2 \cos x)^{m-2}+\frac{m(m-3)}{1 \cdot 2}(2 \cos x)^{m-4}+\ldots$

$\cos x+\binom {n}{1} \cos 2 x+\ldots +\binom {n}{n} \cos (n+1) x$;

$\sin x+\binom {n}{1} \sin 2 x+\ldots +\binom {n}{n} \sin (n+1) x$;

$\sin^{2} x+\sin^{2} 3 x+\ldots +\sin^{2}(2 n-1) x$;

$\cos x+2 \cos 2 x+3 \cos 3 x+\ldots +n \cos n x$;

$\sin x+2 \sin 2 x+3 \sin 3 x+\ldots +n \sin n x$.

$\cos^{2} x+\cos^{2} 2 x+\ldots +\cos^{2} n x=\frac{n}{2}+\frac{\cos (n+1) x \sin n x}{2 \sin x}$;

$\sin^{2} x+\sin^{2} 2 x+\ldots +\sin^{2} n x=\frac{n}{2}-\frac{\cos (n+1) x \sin n x}{2 \sin x}$.