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Prove that sin^(-1) cos (sin^(-1) x) + c...

Prove that `sin^(-1) cos (sin^(-1) x) + cos^(-1) x = (pi)/(2), |x| le 1`

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Case : `0 lt x lt 1`
`L.H.S. = sin^(-1) (cos (sin^(-1) x)) + cos^(-1) (sin (cos^(-1) x))`
`= sin^(-1) (cos (cos^(-1) (sqrt(1 - x^(2))))) + cos^(-1) (sin (sin^(-1) (sqrt(1 -x^(2)))))`
`= sin^(-1) (sqrt(1- x^(2)))) + cos^(-1) (sqrt(1 - x^(2)))`
`= pi//2`
CaseII: `-1 lt x lt 0`
Put `x = -y`, where `y gt 0`
`L.H.S. = sin^(-1) (cos (sin^(-1) (-y))) + cos^(-1) (sin (cos^(-1) (-y)))`
`= sin^(-1) (cos (-sin^(-1) y)) + cos^(-1) (sin (pi - cos^(-1) y))`
`= sin^(-1) (cos (sin^(-1) (y)) + cos^(-1) (sin (cos^(-1) (y))`
`= pi//2` (From case I)
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Knowledge Check

  • If sin^-1 x - cos^-1 x = pi/6 then x is

    A
    44198
    B
    `sqrt3/2`
    C
    -0.5
    D
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