If `y=sin^(-1)x+sin^(-1)sqrt(1-x^2)` , find `(dy)/(dx)` in each of the following cases: (i) `x in (0,\ 1)` (ii) `x in (-1,\ 0)`

Explanation

Given that,
`y=sin^(-1)x+sin^(-1)sqrt(1-x^2)`
Let `x = sintheta`, then `sin^-1x = theta`
Then, our given equation becomes,
`y = sin^-1(sintheta)+sin^-1(sqrt(1-sin^2theta))`
`y = sin^-1(sin theta)+sin^-1(costheta)`

`(i) 0 lt x lt 1`
`=>0 lt sin theta lt 1`
`=>sin^-1 0 lt theta lt sin^-1 (1)`
`=>0 lt theta lt pi/2`
`y = sin^-1(sin theta)+sin^-1(sin(pi/2-theta))`
`y = theta+pi/2 -theta`
`y = pi/2`
`:. dy/dx = 0`


`(ii) -1 lt x lt 0`
`=>-1 lt sin theta lt 0`
`=>-pi/2 lt theta lt 0`
`y = sin^-1(sin theta)+sin^-1(sin(pi/2+theta))`
`y = theta+pi/2 +theta`
`y = pi/2+2theta`
`=>y=pi/2 +2 sin^-1x`
`:. dy/dx = 2/sqrt(1-x^2)`