so f(x) = 2sin(t)cos(t)
it has odd symertry, -f(x) = f(-x). but i dont know how relevent this is
I presume you know the basic formulas?
The coefficient of "cos(nx)" is $\displaystyle a_n= \frac{1}{\pi}\int_{-\pi}^\pi (2 sin(t)cos(t))cos(nt)dt$
The coefficient of "sin(nx)" is $\displaystyle b_n= \frac{1}{\pi}\int_{-\pi}^\pi (2sin(t)cos(t) sin(nt)dt$
Because 2sin(t)cos(t) is an odd function, its Fourier series includes only odd terms- that is no cos(nx) term. [itex]a_n= 0[/itex] for all n.