Question

Write the following in the simplest form:

`sin^-1{(x+sqrt(1-x^2))/sqrt2},-1<x<1`

Answer 1

Let x = sin θ

Now,

`sin^-1{(x+sqrt(1-x^2))/sqrt2}=sin^-1{(sintheta+sqrt(1-sin^2theta))/sqrt2}`

`=sin^-1{(sintheta+costheta)/sqrt2}`

`=sin^-1{1/sqrt2sintheta+1/sqrt2costheta}`

`=sin^-1{cos  pi/4sintheta+sin  pi/4costheta}`

`=sin^-1{sin(theta+pi/4)}`

`=theta+pi/4`

`=pi/4+sin^-1x`

`thereforesin^-1{(x+sqrt(1-x^2))/sqrt2}=cos^-1x+pi/4`