Reflection formula for Euler $\Gamma$ function

The purpose of this problem is to prove the following formula
$$\boxed{\forall z\in]0,1[, \Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}}$$where $\displaystyle\Gamma(z):=\int_0^{+\infty}t^{z-1}\,e^{-t}\,\mathrm{d}t$ is the Euler function. This formula, generalizable to the whole band $0<\Re(z)<1$, is very useful in order to understand how the Riemann $\zeta$-function behaves for $\Re(z)<1$.

We start by some preliminaries.

1) a) Show that $\Gamma(x)$ is defined for all $x>0$. Show also that the function $B$ such that
$$\displaystyle B(x,y):=\int_0^1 t^{x-1}\,(1-t)^{y-1}\,\mathrm{d}t,$$is defined for all $x,y>0$.

1) b) For $x,y>0$, show that
$$\Gamma(x)\Gamma(y)=\Gamma(x+y)B(x,y).$$

1) c) Let $\tilde{f_{\alpha}}\colon x\in [-\pi,\pi[\longmapsto \cos(\alpha x)$ where $\alpha\in\R\backslash \Z$ and let $f_{\alpha}\colon \R\to \R$ be the unique $2\pi$-periodic function such that $f_{\alpha}(x)=\tilde{f_{\alpha}}(x), x\in[-\pi,\pi[$ and $f_{\alpha}$ is even.
Show that $f_{\alpha}$ can be decomposed as a real Fourier series, determine the coefficients of this decomposition and deduce the relation
$$\frac{\pi}{\sin(\alpha \pi)}=\sum_{k=-\infty}\frac{(-1)^k}{k+\alpha}$$

Now we can move to the heart of the proof. To do so, define the integral for all $z\in]0,1[$
$$I_{z}:=\int_0^{+\infty}\frac{t^{z-1}}{1+t}\,\mathrm{d}t$$

2) Show that $I_{z}$ is well-defined for all $0<z<1$.
3) Show that
$$I_z=\int_0^1 \frac{t^{z-1}+t^{-z}}{1+t}\,\mathrm{d}t=\sum_{k=-\infty}^{+\infty}\frac{(-1)^k}{k+z}=\frac{\pi}{\sin(\pi z)}$$4) Show also that
$$\forall x,y>0, B(x,y)=\int_0^{+\infty}\frac{u^{x-1}}{(1+u)^{x+y}}\,\mathrm{d}u$$and conclude.