Differentiate `sin^(-1)sqrt(1-x^2)` with respect to `cos^(-1)x` , if `x in (-1/(2sqrt(2)),\ 1/(2sqrt(2)))`

Explanation

Given that,
`u=sin^(-1)sqrt(1-x^2)`
`v=cos^(-1)x`
Let `x=cos theta`
so, `u=sin^-1 sqrt(1-cos^2theta)`
`=>u=sin^-1 sqrt(sin^2theta)`
`=>u=sin^-1(sintheta)`
Again `v=cos^(-1)x`
`=>v=cos^-1(cos theta)`
`x in (-1/(2sqrt(2)),\ 1/(2sqrt(2)))`
`=>cos theta in (-1/(2sqrt(2)),\ 1/(2sqrt(2)))`
`=>sqrt7/(3sqrt2) lt sin theta lt 1`
so,`=>u=theta`
`(du)/(d theta)=1`
`=>v=theta`
`=>(dv)/(d theta)=1`
`=>(du)/(dv)=1`