Proving orthogonality How to prove a cosine signal with irrational period and another cosine signal with a rational period are orthogonal?

For example, $\cos(4t)$ and $\cos(4\pi t)$ sum is aperiodic. Hence it is said that these two signals are orthogonal over infinite limits? But how to prove this? Normal integration of product of the two signals cannot be used because we get $\cos(\infty)$ which is unknown.
Sinusoids are power signals, and in the case of power signals, cross-correlation between two signals $x(t)$ and $y(t)$ is defined by

$$R_{xy}(t)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}x(\tau)y(t+\tau)d\tau\tag{1}$$

If $x(t)=\cos(\omega_1 t)$ and $y(t)=\cos(\omega_2 t)$ then for $\omega_1\neq\omega_2$ the cross-correlation at $t=0$ is

$$\begin{align}R_{xy}(0)&amp;=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}\cos(\omega_1 t)\cos(\omega_2 t)dt\\
&amp;=\lim_{T\rightarrow\infty}\frac{1}{2T}\left[\frac{\sin((\omega_1-\omega_2)T)}{\omega_1-\omega_2}+\frac{\sin((\omega_1+\omega_2)T)}{\omega_1+\omega_2}\right]\\&amp;=0
\end{align}$$

Note that the definition (1) avoids all convergence problems that you would encounter otherwise.