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Question
Solve `cos^-1sqrt3x+cos^-1x=pi/2`
Solution
`cos^-1sqrt3x+cos^-1x=pi/2`
⇒`cos^-1[sqrt3x xx x-sqrt(1-(sqrt3x)^2)sqrt(1-x^2)]=pi/2` `[becausecos^-1x+cos^-1y=cos^-1(xy-sqrt(1-x^2)sqrt(1-y^2)]`
⇒ `cos^-1[sqrt3x^2-sqrt(1-3x^2)sqrt(1-x^2)]=pi/2`
⇒ `sqrt3x^2=sqrt(1-3x^2)sqrt(1-x^2)=cos pi/2`
⇒ `sqrt3x^2=sqrt(1-3x^2)sqrt(1-x^2)`
⇒ `3x^4=(1-3x^2)(1-x^2)`
⇒ `3x^4=1-3x^2+3x^4-x^2`
⇒ `4x^2=1`
⇒ `x^2=1/4`
⇒ `x=+-1/2`
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