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Prove that : cos^(-1) x + cos^(-1) ((x)/...

Prove that : `cos^(-1) x + cos^(-1) ((x)/(2) + (sqrt( 3-3x^2) )/( 2) ) = (pi)/ (3)`

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Verified by Experts

The correct Answer is:
`x in [(1)/(2), 1]`
`cos^(-1) x + cos^(-1) [(x)/(2) + (1)/(2) sqrt(3 -3x^(2))] = (pi)/(3)`
`rArr cos^(-1) x + cos^(-1) [(1)/(2) x + (sqrt3)/(2) sqrt(1 -x^(2))] = (pi)/(3)`
`rArr cos^(-1) x + cos^(-1).(1)/(2) - cos^(-1) x = (pi)/(3)`
Above holds when `cos^(-1).(1)/(2) ge cos^(-1) x`
`:. x ge (1)/(2)`
`:. x in [(1)/(2), 1]`
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