calculus - Fourier series of: $[log(sin x)]^2$

What is the Fourier expansion of:

$${ \left[ \log\left( \sin x \right) \right] }^{ 2 }$$

This is a well known Fourier series:

$$-\log(\sin x )=\sum_{k=1}^\infty\frac{\cos(2kx)}{k}+\log(2)$$

I couldn't proceed through this. I also tried using: $\sin x=\frac { { e }^{ ix }-{ e }^{ -ix } }{ 2i } $

But still I couldn't get it. Please help.

Is it possible to get a general form of Fourier expansion for:

$${ \left[ \log\left( \sin x \right) \right] }^{ n }$$

Answer

I don't know if it satisfies you, but using $\log \sin x$ expansion I found the follwing: $$
(\ln\sin x)^2=\ln^2 2-\left (x-\frac{\pi}{2}\right )^2 + \left (x-\frac{\pi}{2}\right )\sin x-2\ln2\cos x+\sum_{k=2}^{\infty}2\left (\frac{(k-2)!}{k!}-\frac{\ln2}{k}\right )\cos2kx+\left (x-\frac{\pi}{2}\right )\frac{\sin2kx}{k}
$$
If you like it I will give you details to find it (and, with a strong effort, also the formulation for any power).