First slide

Properties of ITF

Question

Which of the following is the solution set of the equation $2 \cos^{- 1} x = \cot^{- 1} (\frac{2 x^{2} - 1}{2 x \sqrt{1 - x^{2}}}) ?$

Moderate

A

$(0, 1)$
B

$(- 1, 1) - {0}$
C

$(- 1, 0)$
D

$[- 1, 1]$

Solution

$2 \cos^{- 1} x = \cot^{- 1} (\frac{2 x^{2} - 1}{2 x \sqrt{1 - x^{2}}})$
Put $x = \cos θ$ ; L.H.S. = $2 θ$ where    $0 \leq θ \leq π$ and $- 1 \leq x \leq 1$ ……… (1)
R.H.S. = $\cot^{- 1} (\frac{\cos 2 θ}{2 \cos θ | \sin θ |}) = \cot^{- 1} (\cot 2 θ) = 2 θ$ if $0 < 2 θ < π$      …………. (2)
From (1) and (2), we get       $0 < θ < π / 2$
   $∴ x \in (0, 1)$

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