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Question
The greatest and least values of (sin–1x)2 + (cos–1x)2 are respectively ______.
Options
`(5pi^2)/4` and `pi^2/8`
`pi/2` and `(-pi)/2`
`pi^2/4` ad `(-pi^2)/4`
`pi^2/4` and 0
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Solution
The greatest and least values of (sin–1x)2 + (cos–1x)2 are respectively `(5pi^2)/4` and `pi^2/8`.
Explanation:
We have (sin–1x)2 + (cos–1x)2
= (sin–1x + cos–1x)2 – 2 sin–1x cos–1x
= `pi^2/4 - 2sin^1x (pi/2 - sin^-1x)`
= `pi^2/4 - pi sin^-1x + 2(sin^-1x)^2`
= `2[(sin^-1x)^2 - pi/2 sin^-1x + pi^2/8]`
= `2[(sin^-1x - pi/4)^2 + pi^2/16]`
Thus, the least value is `2(pi^2/16)`
i.e. `pi^2/8` and the Greatest value is `2[((-pi)/2 - pi/4)^2 + pi^2/16]`
i.e. `(5pi^2)/4`.
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