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Q.
Range of the function $f ( x )=\cot ^{-1}\{- x \}+\sin ^{-1}\{ x \}+\cos ^{-1}\{ x \}$ where \{\} denotes fractional part function, is
Inverse Trigonometric Functions
Solution:
$ \Theta f ( x )=\frac{\pi}{2}+\cot ^{-1}\{- x \} $
$\Theta 0 \leq\{- x \}<1$
$\Rightarrow \frac{\pi}{4}<\cot ^{-1}\{- x \} \leq \frac{\pi}{2} $
$\Rightarrow \frac{3 \pi}{4}< f ( x ) \leq \pi$