CTFT of a cosine
$ \begin{align} X(f) &=\mathcal{X}(2\pi f) \\ &=\pi \left[\delta (2\pi f- \omega_0) + \delta (2\pi f+ \omega_0)\right] \\ &=\frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \end{align} $
$ Since\ k\delta (kt)=\delta (t),\forall k\ne 0 $
