Let $\psi = A(t)\cos(\theta_1(t))$ and $\phi = B(t)\cos(\theta_2(t))$ two independent waves which phases and amplitudes depend on the time. Then it follows that the intensity of the superposition of these waves is,
$$I \propto (\psi + \phi)^2 = A^2(t)\cos^2(\theta_1(t)) + B^2(t)\cos^2(\theta_2(t)) \\ + 2A(t)B(t)\cos(\theta_1(t))\cos(\theta_2(t)) $$
However, when it comes about taking average over time, for the case of an incoherent superposition, the cross term is zero, so
$$\lim \limits_{T \to \infty}\frac{1}{T} \int_{0}^{T} A(t)B(t)\cos(\theta_1(t))\cos(\theta_2(t)) = 0$$
$$\lim \limits_{T \to \infty}\frac{1}{T} \int_{0}^{T} A^2(t)\cos^2(\theta_1(t)) = \frac{1}{2}\langle A^2(t) \rangle $$
$$\lim \limits_{T \to \infty}\frac{1}{T} \int_{0}^{T} B^2(t)\cos^2(\theta_2(t)) = \frac{1}{2}\langle B^2(t) \rangle $$ and $$I =\frac{1}{2}\langle A^2(t) \rangle + \frac{1}{2}\langle B^2(t) \rangle = I_{\psi} + I_{\phi}$$
How can I mathematically proof these integrals?
NOTE: My aim is to represent an incoherent addition of waves (e.g two light beams partially polarized). If there is any consideration or assumption that I am missing, please, let me know.