I ran into some problems while doing an exercise. The problem goes as follows:
Suppose we have two random independent variables $X$ and $Y$. Both are distributed normally with parameters $(0, \sigma^2)$. $\mathbb{P}(dx)=\frac{1}{\sigma\sqrt{2\pi}} \exp^{-\frac{x}{2\sigma^2}}dx$. For $\gamma \in \mathbb{R}$, we set $U = X \cos\gamma - Y\sin\gamma$ and $V = X \sin\gamma + Y\cos\gamma$. Show that $U$ and $V$ are independent, calculate their distribution function.
What I've tried:
I know that to check the independence I need to use $\mathbb{E}(\varphi (X) \psi (Y) )= \mathbb{E}(\varphi(X)) \cdot \mathbb{E}(\psi(Y)) $ For that I need to calculate $\mathbb{P}_U$, $\mathbb{P}_V$ and $\mathbb{P}_{(U,V)}$. There are two ways to do that, either pushforward measure or density function. So I'm stuck at calculating $\mathbb{P}_U$ since for pushforward measure I can't express $X$ and $Y$ by using only $U$ or $V$. And for density function I have a problem with trigonometric functions since it changes the sign according to the quadrant and so does an inequality $\mathbb{P}(X \cos\gamma - Y\sin\gamma\leq t)$.
Thanks in advance