Question

If $x,y,z\in \left[\begin{array}{cc}-1,& 1\end{array}\right]$ such that ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=0$, find $x+y+z$.

Answer 1

$\begin{array}{c}{\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=0\\ \Rightarrow \therefore {\cos }^{-1}x=0\\ x=1\\ \Rightarrow {\cos }^{-1}y=0\\ \therefore y=1\\ \Rightarrow {\cos }^{-1}z=0\\ z=1\\ \therefore x+y+z=3\end{array}$