Define the following independent random variables $$Y= cos(\theta_1)- cos(\theta_2)$$
Assume $$\theta_1, \theta_2 \sim U[-\pi,\pi]$$ then with convolution we can get $$Y\sim f_Y(y)$$
Now assume that I want to get expected value of function of Y
$$\mathbb{E}[ \frac{1}{G(Y)}]$$
Is the following true $$\mathbb{E}[ \frac{1}{G(Y)}]=\int_{-2}^{2} \frac{1}{G(y)} f_Y(y)= \int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{1}{4\pi^2}\frac{1}{G(cos\theta_1-cos\theta_2)} d\theta_1d\theta_2 $$
thanks