Fourier Transform: $\omega$ vs $f$ as frequency variable
You have the obvious relationship $\omega=2\pi f$. So going from a Fourier transform expressed in terms of $\omega$ to a Fourier transform expressed in terms of $f$ is just a matter of substitution.
The problem you've encountered is very likely in the argument of the Dirac delta impulse. Clearly, $\delta(\omega)=\delta(2\pi f)$, but you need to know the following relationship in order to obtain an expression in terms of $\delta(f)$:
$$\delta(ay)=\frac{1}{|a|}\delta(y),\qquad a\neq 0\tag{1}$$
So we have
$$\delta(\omega)=\delta(2\pi f)=\frac{1}{2\pi}\delta(f)\tag{2}$$
E.g., the Fourier transform of $x(t)=e^{j\omega_0t}=e^{j2\pi f_0t}$ is given by
$$\begin{align}\mathcal{F}\big\{e^{j\omega_0t}\big\}&amp;=2\pi\delta(\omega-\omega_0)\\&amp;=2\pi\delta\big(2\pi (f-f_0)\big)\\&amp;=\delta(f-f_0)\tag{3}\end{align}$$
I'm sure that from this you can derive the corresponding expressions for $x(t)=\cos(\omega_0t)$.