If `x in (1/(sqrt(2)),\ 1)` , differentiate `tan^(-1)((sqrt(1-x^2))/x)` with respect to `cos^(-1)(2xsqrt(1-x^2))` .

Explanation

Given that,
`u=tan^(-1)((sqrt(1-x^2))/x)`
taking `x=cos theta`
`=>u=tan^(-1)((sqrt(1-cos^2 theta))/cos theta)`
`=>u=tan^(-1)(sin theta/cos theta)`
`=>u=tan^-1(tan theta)`
Again `v=cos^(-1)(2xsqrt(1-x^2))`
`=>v=cos^(-1)(2 cos theta sqrt(1-cos^2 theta))`
`=>v=cos^-1(2sin theta * cos theta )`
`=>v= cos^-1 (sin 2 theta )`
`=>v=cos^-1 [cos(pi/2-2 theta)]`
`x in (1/(sqrt(2)),\ 1)`
`=>cos theta in (1/(sqrt(2)),\ 1)`
`=> theta in (0,pi/4)`
`=>2 theta in (0,pi/2)`
`=>pi/2- 2 theta in (0,pi/2)`
`u=theta`
`=>(du)/(d theta)=1`
`v=pi/2-2 theta`
`=>(dv)/(d theta)=-2`
`=>(du)/(dv)= -1/2`