How to relate the distributions of these trigonometric functions of uniform variables? 
$\theta_1,\theta_2$ are two independent random variables distributed uniformly in $[0,2π)$. Let $X=\cos\theta_1,Y=\cos\theta_2.$ Prove that $\frac{X+Y}{2}$ and $XY$ are equal in distribution,$\frac{X+Y}{2}{\buildrel d \over =}XY$.

I can calculate the cdf of $X,Y$, like PDF of cosine of a uniform random variable, but it's difficult to prove the distribution equality: the distribution function is too hard to calculate.
 A: We can avoid calculating distributions altogether.
In the following I will use "$\sim$" to mean "has the same distribution as" while retaining "$=$" to mean strict equality of random variables, as usual.
The cosine is a periodic function with period $2\pi.$  A periodic function $f$ with period $\tau$ simply is one that satisfies $f(x+\tau)=f(x)$ for all $x.$  Equivalently, $f$ is a function of the numbers modulo $\tau$, meaning that before applying $f$ to any number $x,$ $x$ is replaced by its remainder after division by $\tau.$
Periodic functions of a uniform random variable $\Theta$ supported on $[0,\tau)$ generally enjoy some useful properties we might exploit to simplify our work.  These include


*

*$f(\Theta+\omega) \sim f(\Theta)$ for fixed $\omega.$

*$f(-\Theta) \sim f(\Theta).$

*When $\Theta_1$ and $\Theta_2$ are independent uniform random variables, $(1)$ implies $f(\Theta_1+\Theta_2) \sim f(\Theta_1)$ and then $(2)$ implies $f(\Theta_1-\Theta_2)\sim f(\Theta_1).$ 

*Moreover, $\Theta_1\pm\Theta_2$ are independent when computed modulo $\tau$ and therefore the $f(\Theta_1\pm \Theta_2)$ are independent mod $\tau,$ too.
I leave the (easy) demonstrations to you, since this is a self-study problem.
Specializing to $\tau=2\pi$ and $f=\cos,$ use these facts and the rules of trigonometry to justify the following reasoning:
$$\eqalign {
XY &= \cos(\Theta_1)\cos(\Theta_2) \\
&= \frac{1}{2}\left(\cos(\Theta_1-\Theta_2)+\cos(\Theta_1+\Theta_2)\right)\\
&\sim\frac{1}{2}(X+Y)
.}$$ 
