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प्रश्न
Write the following in the simplest form:
`sin^-1{(x+sqrt(1-x^2))/sqrt2},-1<x<1`
उत्तर
Let x = sin θ
Now,
`sin^-1{(x+sqrt(1-x^2))/sqrt2}=sin^-1{(sintheta+sqrt(1-sin^2theta))/sqrt2}`
`=sin^-1{(sintheta+costheta)/sqrt2}`
`=sin^-1{1/sqrt2sintheta+1/sqrt2costheta}`
`=sin^-1{cos pi/4sintheta+sin pi/4costheta}`
`=sin^-1{sin(theta+pi/4)}`
`=theta+pi/4`
`=pi/4+sin^-1x`
`thereforesin^-1{(x+sqrt(1-x^2))/sqrt2}=cos^-1x+pi/4`
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