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If sin^-1x + sin^-1y + sin^-1z =pi then ...

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

Text Solution

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Let `sin^(-1)x=A,sin^(-1)y=B,sin^(-1)z=c`
`x=sinA,y=sinB,z=sinc`
`A+B+C=pi`
`sin2A+sin2B+sin2C=9sinAsinBsinC`
LHS
`2sin(A+B)*cos(A-B)+sin2C`
`2sin(pi-C)*cos(A-B)+2sinCcosC`
`2sinC(cos(A-B)+cosC)`
...
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