sin(a +- b) = sin(a)*cos(b) +- sin(b)*cos(a)
cos(a +- b) = cos(a)*cos(b) -+ sin(a)*sin(b)

c = 1/(2*pi) * integral from pi to 2*pi cos(t)*exp(i*k*(2*pi/2*pi)*t)
  = 1/(2*pi) * integral from pi to 2*pi cos(t)*(cos(k*t) + i*sin(k*t))

cos(t)*cos(n*t) = 1/2 * (cos((1 + k)*t) + cos((1 - k)*t))
cos(t)*sin(n*t) = 1/2 * (sin((1 + k)*t) - sin((1 - k)*t))

c = 1/(4*pi) * integral from pi to 2*pi cos((1 + k)*t) + cos((1 - k)*t) + i*(sin((1 + k)*t) - sin((1 - k)*t))
Sonderfall k = -1:
c = 1/(4*pi) * integral from pi to 2*pi 1 + cos((1 - k)*t) - i*sin((1 - k)*t))
  = 1/(4*pi) * interval from pi to 2*pi t + exp(i*(1 - k)*t)/(i*(1 - k))
  = 1/4 + (1 - (-1)^((1 - k)/2)/i*(1 - k)))
Sonderfall k = 1:
c = 1/4 + (1 - (-1)^((1 + k)/2)/i*(1 + k))) geraten
Sonst:
Keine Lust :D