Question

Write the following in the simplest form:

`sin^-1{(sqrt(1+x)+sqrt(1-x))/2},0<x<1`

Answer 1

Let x = cos θ

Now,

`sin^-1{(sqrt(1+x)+sqrt(1-x))/2} = sin^-1  {(sqrt(1+costheta)+sqrt(1-costheta))/2}`

`=sin^-1{(sqrt(2cos^2  theta/2)+sqrt(2sin^2  theta/2))/2}`

`=sin^-1{(cos  theta/2+sin  theta/2)/sqrt2}`

`=sin^-1{1/sqrt2sin  theta/2+1/sqrt2cos  theta/2}`

`=sin^-1{sin(theta/2+pi/4)}`

`=theta/2+pi/4`

`=(cos^-1x)/2+pi/4`

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