If $a , b , c$ are real numbers $> 1$, then show that $$\frac { 1 } { 1 + \log _ { a ^ { 2 } b } \frac { c } { a } } + \frac { 1 } { 1 + \log _ { b ^ { 2 } c } \frac { a } { b } } + \frac { 1 } { 1 + \log _ { c ^ { 2 } a } \frac { b } { c } } = 3$$

Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Deduce that
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \frac { \pi } { \sin ( \pi x ) }$$

Recall that $x$ is a fixed element of $]0;1[$. Show that:
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \int _ { 0 } ^ { 1 } \left( \frac { t ^ { x - 1 } } { 1 + t } + \frac { t ^ { - x } } { 1 + t } \right) \mathrm { d } t$$

Let t be a real number. The equalities
$$\begin{aligned} & x = e ^ { 2 \cos t } \\ & y = e ^ { 3 \sin t } \end{aligned}$$
are given.
Accordingly, which of the following gives the relationship between $x$ and y that is satisfied for every real number t?
A) $\ln ^ { 2 } x + \ln ^ { 2 } y = 4$
B) $\ln ^ { 2 } x + \ln ^ { 2 } y = 9$
C) $9 \ln ^ { 2 } x + 2 \ln ^ { 2 } y = 27$
D) $\ln ^ { 2 } x + 4 \ln ^ { 2 } y = 28$
E) $9 \ln ^ { 2 } x + 4 \ln ^ { 2 } y = 36$