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Question
If cos–1x > sin–1x, then ______.
Options
`1/sqrt(2) < x ≤ 1`
`0 ≤ x < 1/2`
`-1 ≤ x < 1/2`
x > 0
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Solution
If cos–1x > sin–1x, then `-1 ≤ x < 1/2`.
Explanation:
Here, given that cos–1x > sin–1x
⇒ `sin[cos^-1x] > x`
⇒ `sin[sin^-1 sqrt(1 - x^2)] > x`
⇒ `sqrt(1 - x^2) > x`
⇒ `x < sqrt(1 - x^2)`
⇒ `x^2 < 1 - x^2`
⇒ `2x^2 < 1`
⇒ `x^2 < 1/2`
⇒ `x < +- 1/sqrt(2)`
We know that – 1 ≤ x ≤ 1
So – 1 ≤ x < `1/sqrt(2)`.
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