How to derive this interesting identity for $\log(\sin(x))$

Question

I saw on SE that:

$$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$

This is an extremely useful identity, as it helps solve:

$$\int_{0}^{\pi} \log(\sin(x)) dx$$

But how is it derived? From Taylor series, power series? How do I get this?

Even if someone can start me off that would be great.

And BTW, $\log\sin$ has elementary primitive: http://es.numberempire.com/integralcalculator.php?function=log%28sin%28x%29%29&var=x. — Martín-Blas Pérez Pinilla, Dec 15 '14 at 08:20
@Martín-BlasPérezPinilla What is obvious to one person is not necessarily obvious to other, that's what this is all about. — hjhjhj57, Dec 15 '14 at 08:33
@Martín-BlasPérezPinilla It's only elementary if you consider the polylogarithm to be elementary. Not many people do, I think. — Daniel R, Dec 15 '14 at 08:54
@DanielR, the formula is so long that I have not seen the polylogarithms ah the end! :$ — Martín-Blas Pérez Pinilla, Dec 15 '14 at 09:10

score 2 · Accepted Answer · answered Dec 15 '14 at 08:25

2

Hint: Use $~\ln(1-t)~=~-\displaystyle\sum_{n=1}^\infty\frac{t^n}n~$ in conjunction with Euler's formula.

answered Dec 15 '14 at 08:25

Lucian

1 Answers1