Question

If $\cos {}^{-1}x+\cos {}^{-1}y+{\cos }^{-1}z=3\pi$ then the value of $xy+yz+zx$ is

• A. $-3$
• B. $1$
• C. $3$
• D. none of these

Answer 1

The correct option is C $3$

If $y=co{s}^{-1}\left(x\right)$ $Rϵ[\pi ,0]$ where$R$ is the range of $y$.

Therefore in the above case, which is only possible if
$co{s}^{-1}\left(x\right)=co{s}^{-1}\left(y\right)=co{s}^{-1}\left(z\right)=\pi$

Hence
$x=y=z=-1$

Therefore $xy+yz+zx$
$=3(-1{)}^2=3$