We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by:
$$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
Show that $\Psi$ is of class $\mathcal{C}^1$ on $\mathbf{R}$, then that for all $x \in \mathbf{R}$,
$$\Psi'(x) = 4\sum_{k=1}^{+\infty} \rho^k \sin(2kx)$$