$\newcommand{\Cov}{\mathrm{Cov}}$

I'd like to work out $\Cov(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.

I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first need the joint density, which I'm stuck on.

I can work out the density of $\cos(3U)$; this is similar to this question. $f_{\cos(3U)}(x) = \frac{1}{\pi\sqrt{1 - x^2}}$, $x \in (-1, 1)$. But that and $f_{\cos(2U)}(x)$ are only (directly) useful for individual expected values.

How do I work out the joint density?

(In this answer, the joint density seems to be $1$. Don't know why though.)

I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.

I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first need the joint density, which I'm stuck on.

I can work out the density of $\cos(3U)$; this is similar to this question. $f_{\cos(3U)}(x) = \frac{1}{\pi\sqrt{1 - x^2}}$, $x \in (-1, 1)$. But that and $f_{\cos(2U)}(x)$ are only (directly) useful for individual expected values.

How do I work out the joint density?

(In this answer, the joint density seems to be $1$. Don't know why though.)