I'm studying book about digital filter by Richard Hamming. And there is exercise to get odd and even expansion of g(x)=x where x is from 0 to $\pi$. I understood even expansion, but can't get into odd expansion: $\pi$ - $2$*(${\rm sin} x$ + $\frac{1}{2}$${\rm sin} 2x$ + $\frac{1}{3}$${\rm sin} 3x$...)
Can anybody explain odd expansion of this function g(x)=x?
Thanks in advance.
UPDATE: Let me be more clear. The exersise is: show that g(x) = x has two expansions (0 < x < $\pi$)
$x$ =$\begin{cases}\pi - 2*(\sin(x)+\sin(2x)/2+\sin(3x)/3+...) \\\pi/2 - 4/\pi*(\cos(x)+\cos(3x)/3^2+\cos(5x)/5^2+...)\end{cases}$
I merely understand cosine expansion, but I don't understand sine expansion...