Let f (x) = cosec ^ - 1 [1 + ^ 2 x], ~where~ [·]denotes the...

Given,

f \left(x\right) = {cosec}^{- 1} \left[1 + \sin^{2}x\right],$~where~\left[\cdot\right]~denotes the greatest integer function.</p><p>Since,~</p><p>$- 1 ≤ \sin x ≤ 1$</p><p>$⇒ 0 $\le \sin^{2}$x ≤ 1$</p><p>$⇒ 1 ≤ 1 + \sin^{2}x ≤ 2$</p><p>$⇒ \left[1 + \sin^{2}x\right] = 1, 1\le \sin^{2}$x < 2$~and

\left[1 + \sin^{2}x\right] = 2, if2 \le 1 + \sin^{2}x < 3$.</p><p>So,</p><p>$f \left(x\right) = {cosec}^{- 1} \left(1\right)

$⇒$f\left(x\right) =\dfrac{π}{2}

And,

f \left(x\right) = {cosec}^{- 1}2$</p><p>$⇒ f\left(x\right) = {cosec}^{- 1}2</p><p>Therefore,~f\left(x\right)$~has two values</p><p>\left(\$\{\dfrac{π}{2}, {cosec}^{- 1}2\right\}$