यदि `cos^(-1)x+cos^(-1)y+cos^(-1)z=pi`, सिद्ध कीजिए कि `x^(2)+y^(2)+z^(2)+2xyz=1`

Explanation

माना `cos^(-1)x=A, cos^(-1)y=B , cos^(-1)z=C`
`implies cos A =x , cos B=y,cos C=z`
`:. A+B+C=piimplies A+B=pi-C`
`implies cos(A+B)=cos(pi-C)`
`implies cosA cos B- sinA sin B=-cosC`
`impliescosA cosB-sinA sinB+cosC=0`
`implies cosA cosB + cosC=sinA sin B`
`implies cosA cosB +cosC=sqrt(1-cos^(2)A)sqrt(1-cos^(2)B)`
`implies xy+z=sqrt(1-x^(2))sqrt(1-y^(2))`
`implies (xy)^(2)+z^(2)+2xyz=(1-x^(2))(1-y^(2))` ( दोनों ओर का वर्ग करने पर )
`implies x^(2)y^(2)+z^(2)+2xyz=1-y^(2)-x^(2)+x^(2)y^(2)`
`implies x^(2)+y^(2)+z^(2) +2xyz=1`