CT Signal:
$ X(t) = 6\cos(2\pit) + 8\sin(4\pit)\, $
$ X(t) = 6\frac{e^{j2\pit}+e^{-j2\pit}}{2} + 8\frac{e^{j4\pit}-e^{-j4\pit}}{2}\, $
$ X(t) = 3e^{j2\pit}+3e^{-j2\pit} + 4e^{j4\pit}-4e^{-j4\pit}\, $
With this expression we can conclude:
$ a_1 = 3\, $
$ a_{-1} = 3\, $
$ a_2 = 4\, $
$ a_{-2} = -4\, $
To calculate $ a_k\, $, we use this equation with $ \omega_0\, $ value as 2:
$ x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\omega_0 t}\, $
Then you will find the value of $ a_k \, $ is zero. -- For any k other than 1, -1, 2, -2
