How does spectral density of a power signal change when it is multiplied by $\cos(2\pi f_ct)$?
$$R_{n}(\tau) = E(n(t)n(t-\tau))$$
$$R_{n_i}(\tau) = \cos(2\pi f_ct)\cos(2\pi f_c(t-\tau))E(n(t)n(t-\tau))=\cos(2\pi f_ct)\cos(2\pi f_c(t-\tau))R_{n}(\tau) $$
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
$$R_n(\tau) = \frac{R_{n_i}(\tau)}{\cos(2\pi f_ct)[ \cos(2\pi f_ct ) \cos(2\pi f_c\tau)+ \sin(2\pi f_ct) \sin(2\pi f_c\tau)]}$$