Question

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

• A. $0$
• B. $1$
• C. $3$
• D. $-3$

Answer 1

The correct option is C $3$

Given, ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=3\pi \pi \Rightarrow {\cos }^{-1}x+{\cos }^{-1}y=3\pi -{\cos }^{-1}z\Rightarrow {\cos }^{-1}(xy-\sqrt{1-x^2}\sqrt{1-y^2})=3\pi -{\cos }^{-1}z\Rightarrow xy-\sqrt{1-x^2}\sqrt{1-y^2}=\cos (3\pi -{\cos }^{-1}z)\Rightarrow xy-\sqrt{1-x^2}\sqrt{1-y^2}=-\cos \left({\cos }^{-1}z\right)\Rightarrow xy-\sqrt{1-x^2}\sqrt{1-y^2}=-z\Rightarrow xy+z=\sqrt{1-x^2}\sqrt{1-y^2}\Rightarrow x^2{y}^2+z^2+2xyz=(1-x^2)(1-y^2)=1-x^2-y^2+x^2{y}^2\Rightarrow x^2+y^2+z^2=1-2xyz$

$\Rightarrow xy+yz+zx=3$