Some trigonometric sums, as $\sum_{k=1}^n\cos{k\alpha}$ and $\sum_{k=1}^n\sin{k\alpha}$ , may be telescoped if the terms are first edited by a suitable goniometric formula (``product formula''). E.g. we may write: $$\sum_{k=1}^n\cos{k\alpha} = \frac{1}{\sin\frac{\alpha}{2}}\sum_{k=1}^n\cos{k\alpha}\sin\frac{\alpha}{2}$$ The product formula $\cos{x}\sin{y} = \frac{1}{2}[\sin(x+y)-\sin(x-y)]$ alters this to $$\sum_{k=1}^n\cos{k\alpha} = \frac{1}{2\sin\frac{\alpha}{2}}\sum_{k=1}^n\left(\sin\frac{(2k+1)\alpha}{2}-\sin\frac{(2k-1)\alpha}{2}\right),$$ or $$\sum_{k=1}^n\cos{k\alpha} =\frac{1}{2\sin\frac{\alpha}{2}}\left(\sin\frac{3\alpha}{2}-\sin\frac{\alpha}{2}+\sin\frac{5\alpha}{2} -\sin\frac{3\alpha}{2}+-\ldots+\sin\frac{(2n+1)\alpha}{2}-\sin\frac{(2n-1)\alpha}{2}\right).$$ After cancelling the opposite numbers we obtain the formula

(1)

(2)

Note. The formulae (1) and (2) are gotten also by adding the left side of the former and $i$ times the left side of the latter and then applying de Moivre identity.

Bibliography

1

. . : . II . ``''. (1970).