It is equivalent to prove random process n(t) is equal to z(t)=x(t)cos(ω_ct)-y(t)sin(ω_ct).

To prove random processes n(t) and z(t) are equal, we need to prove the first and second order moments of n(t) and z(t) are equal.

The first order moment is the mean of n(t) and z(t). It is straightforward to show E{n(t)}=E{z(t)}.

Since random process n(t) is Gaussian and real,
R_n(τ) = E{n(t)n(t-τ)} = δ(τ), where δ(τ) is Dirac function and E{} denotes expectation.
Assume x(t) and y(t) are also real and Gaussian, or
R_x(τ) = E{x(t)x(t-τ)}  = R_y(τ) = E{y(t)y(t-τ)}  = c δ(τ),
and indepedent to each other, R_xy(τ)=E{x(t)y(t-τ)}=0;

R_z(τ) = E{z(t)z(t-τ)} = E{(x(t)cos(ω_ct)-y(t)sin(ω_ct))(x(t)cos(ω_c(t-τ)-y(t)sin(ω_c(t-τ))}
= E{x(t)x(t-τ)cos(ω_ct) cos(ω_c(t-τ)) + y(t)y(t-τ)sin(ω_ct) sin(ω_c(t-τ)) }
= E{x(t)x(t-τ)}cos(ω_ct) cos(ω_c(t-τ)) + E{y(t)y(t-τ)}sin(ω_ct) sin(ω_c(t-τ))
= R_x(τ) cos(ω_ct) cos(ω_c(t-τ)) + R_y(τ) sin(ω_ct) sin(ω_c(t-τ))
= c δ(τ) cos²(ω_ct) + c δ(τ) sin²(ω_ct)
(let c=1)
= δ(τ) = R_n(τ)

Therefore, R_x(τ) = R_y(τ) = R_n(τ)