$t(x)=\prod \limits _{k=1}^{2n}\sin \left (\dfrac{x-x_k}{2}\right )$ a trigonometric polynomial proof

Asked May 19 '22 at 23:10

Active May 19 '22 at 23:21

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Show that, for real $x_1,\ldots ,x_{2n}$,the function:$$t(x)=\prod \limits _{k=1}^{2n}\sin \left (\frac{x-x_k}{2}\right )$$is a trigonometric polynomial$$\frac{1}{2}a_0+\sum \limits _{j=1}^na_j\cos (jx)+b_j\sin (jx)$$with real coefficients $a_j,b_j$.

I was thinking in using the De Moivre's formula, but I can't find how to build the transformation. Any idea is appreciated, thanks.

edited May 19 '22 at 23:21

commie trivial

asked May 19 '22 at 23:10

a dysthemic student

Induction on $n$? – Seewoo Lee May 19 '22 at 23:15
@SeewooLee how could u use it? – a dysthemic student May 19 '22 at 23:31
3

Trigonometric formulas like $\sin \alpha \sin \beta = -\frac{1}{2}(\cos(\alpha + \beta) - \cos(\alpha - \beta))$, $\sin\alpha \cos\beta = \frac{1}{2}(\sin(\alpha + \beta) + \sin (\alpha - beta))$ would help you. – Seewoo Lee May 19 '22 at 23:44

$t(x)=\prod \limits _{k=1}^{2n}\sin \left (\dfrac{x-x_k}{2}\right )$ a trigonometric polynomial proof

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