From the book An Introduction to Copulas (Nelson, 2006) page 55, the distribution function of RVs $X = \cos(\Theta)$ and $Y = \sin(\Theta)$ denoted by $H(x,y)$, where $\Theta$ is a uniform random variable on the interval $[0, 2\pi)$, is
\begin{align} H(x,y) = \begin{cases} \frac{3}{4}-\frac{\cos ^{-1}(x)+\cos ^{-1}(y)}{2 \pi }, & x^2+y^2\leq 1 \\ 1-\frac{\cos ^{-1}(x)+\cos ^{-1}(y)}{\pi }, & x^2+y^2>1\land x\geq 0\land y\geq 0 \\ 1-\frac{\cos ^{-1}(x)}{\pi }, & x^2+y^2>1\land x<0\land y\geq 0 \\ 1-\frac{\cos ^{-1}(y)}{\pi }, & x^2+y^2>1\land x\geq 0\land y<0 \\ 0, &\text{otherwise}. \end{cases} \end{align}
We know that the pdf of $X$ and $Y$ is $h(x,y) = \frac{1}{2\pi}, \text{for}~ x^2 + y^2 =1.$
How can we get the $h(x,y)$ as $\frac{\partial^2H(x,y)}{\partial x \partial y}$? I only get $0$ for each conditional. It seems I miss something.