Proof by Induction that relates to the Jacobian Determinant

Question

Define $f_2$ : $ℝ$ $\to$ $ℝ^2$ by putting $$f_2 (\theta)=(\cos(\theta),\sin(\theta)),$$ and for n $\ge3$ define $f_n: ℝ^{n-1}\toℝ^n$ inductively by setting $$f_n=(\theta_1, \theta_2,\dots,\theta_{n-1})=(\sin(\theta_{n-1})f_{n-1}(\theta_1,\dots,\theta_{n-2}),\cos(\theta_{n-1})),$$ where the right hand side is the vector in $ℝ^n$ with the last coordinate equal to $\cos(\theta_{n-1})$ and first $n-1$ coordinates equal to the coordinates of the vector $f_{n-1}(\theta_1,\dots,\theta_{n-2})$ multiplied by the scalar $\sin(\theta_{n-1})$. Finally define $g_n : ℝ^n \to ℝ^n$ by the formula $$g_n(r,\theta_1,\dots,\theta_{n-1})=rf_n(\theta_1,\dots,\theta_{n-1}).$$ Prove that the Jacobian determinant $J_n$ of $g_n$ satisfies $$J_n=(-1)^nr^{n-1}\prod_{j=2}^{n-1}\sin^{j-1}\theta_j$$ My ideas so far is to maybe expand $g_n$ and $g_{n-1}$ in terms of $f_n$ from their jacobians but im still unsure.