How to time average the product of two waves with distinct periods?

Question

I have a product of two cosine waves, each with a different period.

$$f(x) = \cos(\omega_1t-\delta)\cos(\omega_2t-\delta)$$

I would like to find the time average of this product using:

$$\frac1T \int_0^T \cos(\omega_1t-\delta)cos(\omega_2t-\delta)dt$$

The period of $f(x)$ is $T = \omega_1\omega_2$, but with this limit, I get a computer solution that seems too complicated to be correct. Is there any reason to believe this integral has a palatable solution? If the integral is analytic, please sketch the method of solution.

Answer 1

Hint: You may solve it by using$$\cos A.\cos B=\frac{1}{2}(\cos(A+B)+\cos(A-B))$$

Thus: $$\frac{1}{2T}\int^T_0 \cos((\omega_1+\omega_2)t-2\delta)+cos(\omega_1-\omega_2)t.dt $$

Hope you can solve further?