Consider a signal that is a sum of sinusoids, e.g. x ( t ) =...

Step 1
Consider a general sum of sinusoids
$x(t)=\sum _i{C}_i{f}_i(a_{i}t)$
where $f_i(a_{i}t)=sin(a_{i}t)$ or $f_i(a_{i}t)=cos(a_{i}t)$ . The coefficients $C_i$ are simply constants. For example this sum could take the form
$x(t)=2cos(2t)-3cos(3t)+4cos(4t)-5sin(5t)$ .
Now, substitute this sum into the autocorrelation formula:
$R_{xx}(\tau )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x(t+\tau )x(t)dt$
And realize that each term in the integrand can be one of the following four cases:
${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{C}_i{C}_{j}sin(a_i(t+\tau ))sin(a_{j}t)dt=\frac{C_i{C}_j}{2}cos(a_i\tau )$
${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{C}_i{C}_{j}sin(a_i(t+\tau ))cos(a_{j}t)dt=\frac{C_i{C}_j}{2}sin(a_i\tau )$
${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{C}_i{C}_{j}cos(a_i(t+\tau ))sin(a_{j}t)dt=-\frac{C_i{C}_j}{2}sin(a_i\tau )$
${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{C}_i{C}_{j}cos(a_i(t+\tau ))cos(a_{j}t)dt=\frac{C_i{C}_j}{2}cos(a_i\tau )$
So the general autocorrelation can be written as a double sum:
$R_{xx}(\tau )=\frac{1}{2}\sum _{ij}{C}_i{C}_j{r}_{ij}(\tau )$
where
$r_{ij}(\tau )=\{\begin{array}{ll}cos(a_i\tau )& \text{if }{f}_i(a_{i}t)=sin(a_{i}t)\text{ and }{f}_j(a_{j}t)=sin(a_{j}t)\\ cos(a_i\tau )& \text{if }{f}_i(a_{i}t)=cos(a_{i}t)\text{ and }{f}_j(a_{j}t)=cos(a_{j}t)\\ sin(a_i\tau )& \text{if }{f}_i(a_{i}t)=sin(a_{i}t)\text{ and }{f}_j(a_{j}t)=cos(a_{j}t)\\ -sin(a_i\tau )& \text{if }{f}_i(a_{i}t)=cos(a_{i}t)\text{ and }{f}_j(a_{j}t)=sin(a_{j}t)\end{array}$