1.

`y = sin ^(-1)((1 - x^(2))/(1+ x^(2))) 0 lt x lt 1`

Answer» `y = sin ^(-1)((1-x^(2))/(1+ x^(2)))`
Let x = tantheta
`rArr theta^(-1) x`
`rArr y = sin ^(-1)((1- tan^(2)theta)/(1+ tan^(2)theta))`
`= sin ^(-1)((cos ^(2)theta-sin^(2)theta)/(cos^(2)theta+ sin^(2)theta))`
`= sin ^(-1)(cos 2 theta) = sin ^(-1) sin((pi)/(2)-2 theta)`
`= (pi)/(2) - 2 theta= (pi)/(2)-2tan^(-1) x`.
`rArr (dy)/(dx)=(d)/(dx)((pi)/(2)-2tan^(-1)x.)`
`= 0 - (2xx1)/(1+x^(2))=-(2)/(1+x^(2))`.


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