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Question
Write the following in the simplest form:
`sin^-1{(sqrt(1+x)+sqrt(1-x))/2},0<x<1`
Solution
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`
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