The range of $s i n^{- 1} x - c o s^{- 1} x$ is

[\frac{- 3 π}{2}, \frac{π}{2}]

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[\frac{- 5 π}{2}, \frac{π}{3}]

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[\frac{- 3 π}{2}, π]

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[0, \frac{π}{2}]

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Solution

The correct option is A $[\frac{- 3 π}{2}, \frac{π}{2}]$
$s i n^{- 1} - c o s^{- 1} x = \frac{π}{2} - c o s^{- 1} x - c o s^{- 1} x = \frac{π}{2} - 2 c o s^{- 1} x$
$0 \leq c o s^{- 1} x \leq π \Rightarrow 0 \leq 2 c o s^{- 1} x \leq 2 π \Rightarrow - 2 π \leq - 2 c o s^{- 1} x \leq 0 \Rightarrow \frac{- 3 π}{2} \leq \frac{π}{2} - 2 c o s^{- 1} x \leq \frac{π}{2}$
$∴$ Range $= [\frac{- 3 π}{2}, \frac{π}{2}]$

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