First slide

Properties of ITF

Question

If sum of the elements in the range of $f (x) = \sin^{- 1} x + \cos^{- 1} x + \tan^{- 1} x + \tan^{- 1} \frac{1}{x} + \sec^{- 1} x + {cosec}^{- 1} x is k π, then k =$

Moderate

A

3
B

0
C

1
D

2

Solution

$We know that (1) \sin^{- 1} x + \cos^{- 1} x = \frac{π}{2}; \forall x \in [- 1, 1]$
$(2) \tan^{- 1} x + \tan^{- 1} \frac{1}{x} = \{\begin{array}{cl} \frac{π}{2} & if x \geq 0 \\ - \frac{π}{2} & if x < 0 \end{array}$
$(3) \sec^{- 1} (x) + {cosec}^{- 1} (x) = \frac{π}{2}; \forall x \in (- \infty, - 1] \cup [1, \infty)$
Only -1 and 1 satisfies the given equation
$\begin{array}{l} f (- 1) = \frac{π}{2} + (\frac{- π}{2}) + (\frac{π}{2}) = \frac{π}{2} \\ f (1) = \frac{π}{2} + \frac{π}{2} + \frac{π}{2} = \frac{3 π}{2} \\ ∴ Range is \{\frac{π}{2}, \frac{3 π}{2}\} \end{array}$
$\begin{array}{l} Sum of the elements in the range is 2 π \\ \Rightarrow k = 2 \end{array}$
Therefore, the correct answer is (4).

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