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:
$$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
Deduce that for all $x \in \mathbf{R}$,
$$\Psi(x) = 2\ln\left(\frac{a+b}{2}\right) - 2\sum_{k=1}^{+\infty} \frac{\cos(2kx)}{k}\rho^k$$