$\textbf{Problem:}$ Consider we have a class $C^1$ parameterization $\psi:[a_1,b_1]\times[a_2,b_2]\rightarrow\mathbb{R}^3-\{0\}$ for the surface $S$. Also, consider that $S$ is such that the map $\varphi:S\rightarrow S^2$, with $\varphi(x,y,z)=\lVert(x,y,z)\lVert^{-1}(x,y,z)$, is injective.
PS: $\lVert(x,y,z)\lVert$ is the 2-norm.
The solid angle of $S$ is the area of $\varphi(S)$. I'm asked to show that the solid angle is $$\Bigg|\int_{a_1}^{b_1}\int_{a_2}^{b_2}\frac{1}{\lVert(\psi(s,t))\lVert^3}\psi(s,t)\cdot\big(\psi_s(s,t)\times\psi_t(s,t)\big)dtds\Bigg|. $$
$\textbf{My trial:}$ I started noting that $$\frac{\partial(\varphi\circ\psi)_i}{\partial s}=\frac{1}{\lVert\psi\lVert^3}\Bigg(\lVert\psi\lVert^2\frac{\partial\psi_i}{\partial s}-\psi_i(\psi_s\cdot\psi)\Bigg)$$ for $i=1,2,3$, and analogous for variable $t$. I simplified the notation doing $\psi=\psi(s,t)$, $\psi_s=\bigg(\frac{\partial\psi_1}{\partial s}(s,t),\frac{\partial\psi_2}{\partial s}(s,t),\frac{\partial\psi_3}{\partial s}(s,t)\bigg)$ and $\psi_i$ is the ith coordinate of $\psi(s,t)$.
I can parameterize $\varphi(S)$ using the composite $\varphi\circ\psi$, and the following expression gives the area of the parameterized surface $\varphi(S)$: $$\int_{a_1}^{b_1}\int_{a_2}^{b_2}\Bigg\lVert\Bigg(\frac{\partial(\varphi\circ\psi)_1,}{\partial s}\frac{\partial(\varphi\circ\psi)_2}{\partial s},\frac{\partial(\varphi\circ\psi)_3}{\partial s}\Bigg)\times\Bigg(\frac{\partial(\varphi\circ\psi)_1,}{\partial t}\frac{\partial(\varphi\circ\psi)_2}{\partial t},\frac{\partial(\varphi\circ\psi)_3}{\partial t}\Bigg)\Bigg\lVert dtds $$ I think the absolute value we have is to take care of some orientation issue. Anyway, using the initial relation and doing the calculations in this integral, we get $$\int_{a_1}^{b_1}\int_{a_2}^{b_2}\frac{1}{\lVert\psi\lVert^3}\Big\lVert \lVert\psi\lVert(\psi_s\times\psi_t)-\lVert\psi\lVert^{-1}\Big((\psi_t\cdot\psi)(\psi_s\times\psi)-(\psi_s\cdot\psi)(\psi_t\times\psi)\Big)\Big\lVert dtds= $$ $$=\int_{a_1}^{b_1}\int_{a_2}^{b_2}\frac{1}{\lVert\psi\lVert^3}\Big\lVert \lVert\psi\lVert(\psi_s\times\psi_t)-\lVert\psi\lVert^{-1}\Big(\big((\psi_t\cdot\psi)\psi_s-(\psi_s\cdot\psi)\psi_t\big)\times\psi\Big)\Big\lVert dtds $$ I don't know what to do from here, thanks.