If α≤sin−1⁡x+cos−1⁡x+tan−1⁡x≤β, then

A
$α = \frac{π}{4}, β = \frac{3 π}{4}$
B
$α = - π, β = 2 π$
C
$α = 0, β = π$
D
none of these

Solution:

We have,

$\sin^{- 1} x + \cos^{- 1} x = \frac{π}{2}$ for all $x \in [- 1, 1]$

Also,

$- \frac{π}{4} \leq \tan x \leq \frac{π}{4}$ for all $x \in [- 1, 1]$

$\begin{array}{l} ∴ - \frac{π}{4} + \frac{π}{2} \leq \sin^{- 1} x + \cos^{- 1} x + \tan^{- 1} x \leq \frac{π}{4} + \frac{π}{2} \\ \Rightarrow \frac{π}{4} \leq \sin^{- 1} x + \cos^{- 1} x + \tan^{- 1} x \leq \frac{3 π}{4} \end{array}$

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