Question

If −1 < x < 0, then write the value of `sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))`

Answer 1

Let `x=-tany`
Where `0<y< pi/2`
Then,

`sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))=sin^-1((-2tany)/(1+tan^2y))+cos^-1((1-tan^2y)/(1+tan^2y))`

`=sin^-1{-sin(2y)}+cos^-1{cos(2y)}`

`=-sin^-1{sin(2y)}+cos^-1{cos(2y)}`

`=-2y+2y`

= 0

`therefore sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))=0`