Prove that `sin^(-1). ((x + sqrt(1 - x^(2))/(sqrt2)) = sin^(-1) x + (pi)/(4)`, where `- (1)/(sqrt2) lt x lt(1)/(sqrt2)`

Answer

`sin^(-1) x + (pi)/(4)`

Explanation

Let `x = sin theta`, where
`-(1)/(sqrt2) lt x lt (1)/(sqrt2)`
`rArr - (pi)/(4) lt theta lt (pi)/(4)`
`sin^(-1) ((x + sqrt(1 - x^(2)))/(sqrt2)) = sin.^(-1) ((sin theta + cos theta)/(sqrt2))`
`= sin.^(-1) (sin (theta + (pi)/(4)))`
`= theta + (pi)/(4) ( :' theta + (pi)/(4) in (0, (pi)/(2)))`
`= sin^(-1) x + (pi)/(4)`