First slide

Properties of ITF

Question

If a ≤ tan^–1x + cot^–1x + sin^–1x ≤ b, then

Moderate

A

a = 0
B

$b = \frac{π}{2}$
C

$a = \frac{π}{4}$
D

b = π

Solution

We have,
$\tan^{- 1} x + \cot^{- 1} x + \sin^{- 1} x = \frac{π}{2} + \sin^{- 1} x Since - \frac{π}{2} \leq \sin^{- 1} x \leq \frac{π}{2} \Rightarrow 0 \leq \frac{π}{2} + \sin^{- 1} x \leq π \begin{array}{l} \Rightarrow 0 \leq \tan^{- 1} x + \cot^{- 1} x + \sin^{- 1} x \leq π [From (1)] \\ ∴ a = 0 and b = π \end{array}$

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