We propose to calculate the area $\mathscr{A}$ of the domain $\mathscr{H}$ of $\mathbb{R}^2$ containing all the points $w(n, t)$ when $n$ ranges over $\mathbb{N}^*$ and $t$ ranges over $I = \left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$. This domain is bounded by two parametrized arcs defined by $$z : I \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}\theta(t)} = \sqrt{1 + 3\sin^2 t}\, \mathrm{e}^{\mathrm{i}\left(\arctan(2\tan t) + \pi \mathrm{E}\left(\frac{t}{\pi} + \frac{1}{2}\right)\right)}$$ $$v : I \rightarrow \mathbb{C}, t \mapsto \sqrt{1 + 3\sin^2 t}\left(1 + \frac{1}{4}\sin\left(\frac{2}{3}\left(t - \frac{\pi}{4}\right)\right)\right) \mathrm{e}^{\mathrm{i}\left(\arctan(2\tan t) + \pi \mathrm{E}\left(\frac{t}{\pi} + \frac{1}{2}\right)\right)}$$
III.D.1) Recall the statement of the Green-Riemann theorem. Explain how this theorem translates in the case of an area calculation. III.D.2) Recall the formula giving the scalar product of two complex numbers. Deduce the expression of the scalar product $\langle u \circ v(t), v'(t) \rangle$, when $u$ and $v$ are the applications $u : \mathbb{C} \rightarrow \mathbb{C}, z \mapsto \mathrm{i}z$ and $v : t \mapsto \sigma(t) \mathrm{e}^{\mathrm{i}\mu(t)}$, where $\sigma$ and $\mu$ are two functions defined on an interval $J$ of $\mathbb{R}$, with real values and of class $C^1$. III.D.3) If $d(t) = \arctan(2\tan(t))$, simplify $\frac{1}{2}\left(1 + 3\sin^2 t\right) d'(t)$. III.D.4) Deduce from the previous questions an expression of $\mathscr{A}$ in the form of an integral. Simplify this integral using the identity obtained in III.D.3). Finally, calculate $\mathscr{A}$.