Paradox in Theory of probability of transition to polar coordinates

Question

Let there are two independent random variables $X$,$Y$ with normal distribution. Vector $(X, Y)$ can be considered as a random point on the plane. Let $R$ and $\phi$ polar coordinates of this point. assuming that $X =Y $ we get that the distribution $R^2 = 2X^2$ coincides with the distribution of the square of a random quantity multiplied by 2. At the same time, provided that $\phi = \pi/4$ or $\phi = 5\pi/4$ distribution of random variables $R^2 = X^2 + Y^2$ is the same as the distribution of the sum of squares of two independent standard normal values. Therefore, we got different distributions when $X=Y$ and when $\phi = \pi/4$ or $\phi = 5\pi/4$ and it's a paradox. What's the catch?

Answer 1

The catch is that as soon as you assumed that $X=Y$, they are not independent anymore.