grandes-ecoles 2019 Q20

grandes-ecoles · France · centrale-maths2__pc Reduction Formulae Derive a Product or Series Representation from Reduction Formulae

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Show $$\forall n \in \mathbb{N}^{\star}, \forall x \in \mathbb{R}, \quad \sin(\pi x) = \pi x \frac{I_{2n}(x)}{I_{2n}(0)} \prod_{k=1}^{n} \left(1 - \frac{x^2}{k^2}\right)$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Show
$$\forall n \in \mathbb{N}^{\star}, \forall x \in \mathbb{R}, \quad \sin(\pi x) = \pi x \frac{I_{2n}(x)}{I_{2n}(0)} \prod_{k=1}^{n} \left(1 - \frac{x^2}{k^2}\right)$$

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