Question

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

• A. 1
• B. 0
• C. -3
• D. 3

Answer 1

Answer: (D)

${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=3\pi$
We know that $0\le {\cos }^{-1}x\le \pi$
$\Rightarrow {\cos }^{-1}x={\cos }^{-1}y={\cos }^{-1}z=\pi$
$\Rightarrow x=y=z=-1$
Now, $xy+yz+zx$
$=\left(-1\right)\left(-1\right)+\left(-1\right)\left(-1\right)+\left(-1\right)\left(-1\right)=3$