I find an equation from wiki https://en.wikipedia.org/wiki/Solid_angle#cite_note-6, $$\tan{\frac{\Omega}{2}}=\frac{|\vec{a}\cdot(\vec{b}\times\vec{c})|}{|a||b||c|+(\vec{a}\cdot\vec{b})|c|+(\vec{a}\cdot\vec{c})|b|+(\vec{b}\cdot\vec{c})|a|}\quad(1)$$
I want to calculate the solid angle or area spanned by these three vectors $$\vec{a}=(0,0,1),\\ \vec{b}=(\sin{\theta_{b}}\cos{\phi_b},\sin{\theta_b}\sin{\phi_b},\cos{\theta_b}),\\ \vec{c}=(\sin{\theta_{c}}\cos{\phi_c},\sin{\theta_c}\sin{\phi_c},\cos{\theta_c}). $$ Firstly, I can consider a special case $\theta_b=\theta_c=\theta$, $\phi_b=0$, $\phi_c=\pi/2$, then the area can be calculted by $$\int^{\theta}_{0}\sin{\theta}d\theta\int^{\phi_c}_{\phi_b}d\phi=(1-\cos{\theta})(\phi_c-\phi_b)\equiv\Omega_{A}\quad(2)$$ Eq.(1) gives $$\tan{\frac{\Omega}{2}}=\frac{\sin^{2}{\theta}\sin{(\phi_c-\phi_b)}}{(1+\cos{\theta})^{2}+\sin^{2}{\theta}\cos{(\phi_c-\phi_b)}}\quad(3)$$ Take $\phi_c=\pi/2$,$\phi_b=0$, Eq.(2) gives $\Omega_{A}=(1-\cos{\theta})\pi/2$, Eq.(3) gives $\tan{\frac{\Omega}{2}}=\sin^{2}{\theta}/(1+\cos{\theta})^{2}$, $\Omega_{A}=\Omega$ only when $\theta=\pi/2$.
What is wrong here? Why the solid angle calculated by area and the equation from wiki give different result?
I find the loophole in the method eq.(2), since I fix $\theta$ which means bc is not geodesic, so the result from integral is larger than the wiki equation (1).