How does spectral density of a power signal change when it is multiplied by $\cos(2\pi f_ct)$? Suppose we have a power signal $n(t)$ whose power- auto-correlation function is: $R_{n}(\tau)$.

Now we multiply the signal by a cosine as
$$\cos(2\pi f_ct) \cdot n(t)$$
whose auto-correlation is $R_{n_i}(\tau)$. 

How is $R_{n}(\tau)$ mathematically related to $R_{n_i}(\tau)$ ?
$$R_{n}(\tau) = E(n(t)n(t-\tau))$$
$$R_{n_i}(\tau) = \cos(2\pi f_ct)\cos(2\pi f_c(t-\tau))E(n(t)n(t-\tau))=\cos(2\pi f_ct)\cos(2\pi f_c(t-\tau))R_{n}(\tau) $$
Using $\cos(a-b) = \cos a \cos b + \sin a \sin b$, we get
$$R_{n_i}(\tau) = \cos(2\pi f_ct)[ \cos(2\pi f_ct ) \cos(2\pi f_c\tau)+ \sin(2\pi f_ct) \sin(2\pi f_c\tau)]R_{n}(\tau) $$
or
$$R_n(\tau) = \frac{R_{n_i}(\tau)}{\cos(2\pi f_ct)[ \cos(2\pi f_ct ) \cos(2\pi f_c\tau)+ \sin(2\pi f_ct) \sin(2\pi f_c\tau)]}$$