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प्रश्न
Solve the equation for x:sin−1x+sin−1(1−x)=cos−1x
उत्तर
We have,
sin−1x+sin−1(1−x)=cos−1x
`sin^−1 x -cos^−1 x=-sin^−1 (1−x)`
`sin^−1 x -cos^−1 x=sin^−1 (x-1) ......................(1) [because sin^(-1)(-x)=-sin^-1x]`
`Put sin^-1 x=theta and cos^-1 x= phi`
`sin theta=x and cos phi=x`
`then cos theta=sqrt(1-sin^2theta) and sin phi=sqrt(1-cos^2 phi)`
`cos theta=sqrt(1-x^2) and sin phi =sqrt(1-x^2)`
Applying the formula:
`sin(theta-phi)=sin theta cos phi-cos theta sin phi` , we get
`sin(theta-phi)=x.x-sqrt(1-x^2)sqrt(1-x^2)`
`sin(theta-phi)=x^2-(1-x^2)`
`sin(theta-phi)=x^2-1+x^2`
`sin(theta-phi)=2x^2-1`
`(theta-phi)=sin^-1(2x^2-1)`
`sin^-1x - cos^-1 x=sin^-1(2x^2-1).............(2)`
From (1) and (2), we get
`sin^-1 (2x^2-1)= sin^-1 (x-1)`
`2x^2-x=0`
`x(2x-1)=0`
`x=0 or 2x-1=0`
`x=0 or x=1/2`
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