If `cos^(-1)x+cos^(-1)y+cos^(-1)z=pi,t h e n` `x^2+y^2+z^2+2x y z=1` `2(sin^(-1)x+sin^(-1)y+sin^(-1)z)=cos^(-1)x+cos^(-1)y+cos^(-1)z` `x y+y z+z x=x+y+z-1` `(x+1/x)+(y+1/y)+(z+1/z)geq6`

Options

A. `x^(2)+y^(2)=z^(2)`
B. `x^(2)+y^(2)+z^(2)=0`
C. `x^(2)+y^(2)+z^(2)=1-2xyz`
D. None of these

Answer

C

Explanation

Given `cos^(-1)x+cos^(-1)y+cos^(-1)z=pi`
`implies cos^(-1)(xy-sqrt(1-x^(2))sqrt(1-y^(2)))=pi-cos^(-1)z`
`implies xy-sqrt(1-x^(2))sqrt(1-y^(2))=cos (pi-cos^(-1)z)`
`=-cos(cos^(-1z))=0`
`implies xy+z=sqrt(1-x^(2))sqrt(1-y^(2))`
`implies x^(2)y^(2)+z^(2)+2xyz =(1-x^(2))(1-y^(2))`
`=1-x^(2)-y^(2)+x^(2)y^(2)`
`implies x^(2)+y^(2)+z^(2)=1-2xyz`