We define for all real $x > 0$ the sequences $(I_{n}(x))_{n \geqslant 1}$ and $(J_{n}(x))_{n \geqslant 0}$ by: $$I_{n}(x) = \int_{0}^{n} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} dt, \qquad J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Establish Euler's identity: $$\forall x > 0, \quad \Gamma(x) = \lim_{n \rightarrow +\infty} \frac{n! \, n^{x}}{x(x+1) \cdots (x+n)}$$

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 $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Using the result of Q20, deduce $$\forall n \in \mathbb{N}^{\star}, \forall x \in ]0,1[, \quad \cos(\pi x) = \frac{1}{2} \frac{I_{4n}(2x)}{I_{4n}(0)} \frac{I_{2n}(0)}{I_{2n}(x)} \prod_{p=1}^{n} \left(1 - \frac{4x^2}{(2p-1)^2}\right)$$