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If x<0, then prove that cos^(-1)x=pi-sin...

If `x<0,` then prove that `cos^(-1)x=pi-sin^(-1)sqrt(1-x^2)`

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Let `x = cos theta " or " cos^(-1) x = theta`
Since `x lt 0, theta in [pi//2, pi]`
Now, `sin^(-1) sqrt(1 - x^(2)) = sin^(-1) sqrt(1 - cos^(2) theta)`
`= sin^(-1) (sin theta) != theta " " ( :' theta !in [-pi//2, pi//2])`
`= sin^(-1) (sin(pi - theta) = pi - theta`
`rArr cos^(-1) x = pi - sin^(-1) sqrt(1 - x^(2))`
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