Question

If ${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{3\pi }{2}$, then find the minimum value of $x^2+y^2+z^2$

Answer 1

${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{3\pi }{2}$ is only possible when

${\sin }^{-1}x={\sin }^{-1}y={\sin }^{-1}z=\frac{\pi }{2}$

$\Rightarrow x=y=z=1$

$\therefore x^2+y^2+z^2=1+1+1=3$