Prove that: `sin^(-1){(sqrt(1+x)+sqrt(1-x))/2}=pi/4+(sin^(-1)x)/2,""0 < x < 1`

Explanation

`L.H.S. = sin^-1((sqrt(1+x)+sqrt(1-x))/2)`

Let `x = sintheta => theta = sin^-1x`

Then,

`L.H.S. = sin^-1((sqrt(1+sintheta)+sqrt(1-sintheta))/2)`

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

`= sin^-1((sqrt(2cos^2(pi/4-theta/2))+sqrt(2sin^2(pi/4-theta/2)))/2)`

`= sin^-1((sqrt2cos(pi/4-theta/2)+sqrt2sin(pi/4-theta/2))/2)`

`= sin^-1(1/sqrt2cos(pi/4-theta/2)+1/sqrt2sin(pi/4-theta/2))`

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