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Question
If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then ______ < y < ______.
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Solution
If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then – 2π < y < 2π.
Explanation:
y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))`
⇒ y = `2 tan^-1x + 2 tan^-1x`1
⇒ y = `4 tan^-1x` ......`[because sin^1 ((x)/(1 +x^2)) = 2tan^-1x]`
Now `(-pi)/2 < tan^-1x < pi/2`
⇒ `-4 xx pi/2 < 4 tan^-1x < 4 xx pi/2`
⇒ – 2π < y < 2π.
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