If `sin^(-1)x+sin^(-1)y+sin^(-1)z=pi` , prove that: `xsqrt(1-x^2)+ysqrt(1-y^2)+zsqrt(1-z^2)=2x y z`

Options

A. 2xyz
B. `x^(2)+y^(2)+z^(2)`
C. `xy+yz+zx`
D. none of these

Explanation

Let `sin^(-1) x =A sin^(-1) y = B and sin^(-1) z = C` Then `x= sin A, y sin B z = sin C`
Also
`sin^(-1)x+ sin^(-1) y + sin^(-1)z=pi`
`rarr A+B+C=pi`
`rarr sin 2A + sin 2B + sin 2C =4 A sin B sin C`
`rarr sin A cos A + sin B cos B + sin C cos C =2 sin A sin B sin C`
`rarr xsqrt(1-x^(2))+ysqrt(1-y^(2))+zsqrt(1-z^(2))=2xyz`