If `sin^(-1)x+sin^(-1)y+sin^(-1)z=pi,t h e nx^4+y^2+z^4+4x^2y^2z^2=K(x^2y^2+y^2z^2+z^2x^2),` where `K` is equal to 1 (b) 2 (c) 4 (d) none of these

Options

A. 1
B. 2
C. 4
D. none of these

Explanation

We have
`sin^(-1)x+sin^(-1)y+sin^(-1)z=pi`
`rarr sin^(-1)x +sin^(-1)y=pi -sin^(-1)z`
`rarr cos(sin^(-1)x+sin^(-1)y)=cos(pi-sin^(-1)z)`
`rarr cos(sin^(-1)x)cos(sin^(-1)y)-sin^(-1)x)sin(sin^(-1)y)=-cos(sin^(-1)z)`
`rarr sqrt(1-x^(2))sqrt(1-y^(2))-xy=-sqrt(1-z^(2))`
`rarr (1-x^(2))(1-y^(2))=x^(2)y^(2)-2xy sqrt(1-z^(2))` ltrbgt `rarr 1-x^(2)-y^(2)+x^(2)y^(2)=x^(2)y^(2)-2xysqrt(1-z^(2))`
`rarr x^(2)+y^(2)-z^(2)=x^(2)=4x^(2)yu^(2)(1-a^(2))`
`rarr x^(4)+y^(4)+z^(4)+4x^(2)y^(2)z^(2)=2 x^(2)y^(2)z^(2)+y^(2)z^(2)+z^(2)x^(2)`