If `u=sin(mcos^(-1)x),v=cos(msin^(-1)x),p rov et h a t(d u)/(d v)=sqrt((1-u^2)/(1-v^2))`

Explanation

`u = sin(m cos^-1 x) `
`v = cos(m sin^-1 x) `
`sin^-1 u = m cos^-1 x`
and `cos^-1 v = msin^-1 x`
1 + 2
`sin^-1 u + cos^-1 v = m(cos^-1 x + sin^-1 x) = m pi/2`
diff wrt v
`1/(sqrt(1-u^2)) (du)/(dv) + (-1)/sqrt(1-v^2)= 0 `
`(du)/(dv) = sqrt((1-u^2)/(1-v^2))`
Answer