Express as a fourier series or another series for a function

Question

there is Someone asked me a question on a social media , And the question was how to represent $\ln^2\left | \cot \left ( \frac{\Theta x}{2} -\frac{\pi}{4}\right )\right | $ as a series where $\Theta >0 $. When I saw this function I remembered the inverse gudermanian function. And I represented it as a fourier series, which it is $4\sum_{n=1}^\infty \frac{ \sin\left ( \frac{n \pi}{2} \right ) \sin\left ( n \Theta x \right )} n =\ln(\cot\left | \frac{\Theta x } 2 -\frac{\pi}{4} \right |)$ then , I tried to express $\ln^2$. I got $16\sum_{n=1}^\infty \frac{\sin^2 \left ( \frac{n \pi}{2} \right ) \sin^2(n \Theta x)}{n^2}$, But I think the second one is wrong but why? I don't know this all what I can do. I'm looking for any help and I will be hopeful for mathstack team's help