Box-Muller Transform: From Uniforms to Independent Normals

Let $U_1, U_2$ be independent $\operatorname{Uniform}(0,1)$ random variables. Define $$Z_1 = \sqrt{-2\ln U_1}\,\cos(2\pi U_2), \qquad Z_2 = \sqrt{-2\ln U_1}\,\sin(2\pi U_2).$$

(a) Compute the Jacobian of the inverse transformation from $(Z_1, Z_2)$ back to $(U_1, U_2)$.

(b) Show that $Z_1$ and $Z_2$ are independent $N(0,1)$ random variables.