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Sove `2 cos^(-1) x = sin^(-1) (2 x sqrt(1 - x^(2)))` |
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Answer» Let x = cos y, where `0 le y le pi, |x| le 1` `2 cos^(-1) x = sin^(-1) (2 x sqrt(1 -x^(2)))`...(i) `rArr 2 cos^(-1) (cos y) = sin^(-1) (2 cos y sqrt(1 - cos^(2) y))` `= sin^(-1) (2 cos y sin y)` `= sin^(-1) (sin 2 y)` `rArr sin^(-1) (sin 2 y) = 2y " for " -pi//2 le y le pi//4` and `2 cos^(-1) (cos y) = 2y " for " 0 le y le pi` Thus, Eq. (i) holds only when `y in [0, pi//4]` `rArr x in [1//sqrt2, 1]` |
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