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

Answer

B

Explanation

Since `sin^(-1) x + sin^(-1) y + sin^(-1) z = pi`
`:. Sin^(-1) x + sin^(-1) y = pi - sin^(-1) z`
`rArr sin^(-1) (x sqrt(1 - y^(2)) + y sqrt(1 -x^(2))) = pi - sin^(-1) (z)`
`rArr x sqrt(1- y^(2)) + y sqrt(1 -x^(2)) = sin (pi - sin^(-1) (z)) = sin (sin^(-1) z) = z`
or `x^(2) (1 -y^(2)) = z^(2) + y^(2) (1 - x^(2)) - 2zy sqrt(1 -x^(2))`
or `(x^(2) -y^(2) -z^(2))^(2) = 4y^(2) z^(2) (1 -x^(2))`
or `x^(4) + y^(4) + z^(4) - 2x^(2)y^(2) - 2x^(2) z^(2) + 2y^(2) z^(2) = 4y^(2) z^(2) = 4y^(2) z^(2) - 4x^(2) y^(2) z^(2)`
or `x^(4) + y^(4) + z^(4) + 4x^(2) y^(2) z^(2) = 2(x^(2) y^(2) + y^(2) z^(2) + z^(2) x^(2))`
`rArr K = 2`