First slide

Properties of ITF

Question

Which of the following is the solution set of the equation $\sin^{- 1} x = \cos^{- 1} x + \sin^{- 1} (3 x - 1) ?$

Moderate

A

$[0, \frac{1}{3}]$
B

$(\frac{1}{3}, \frac{2}{3}]$
C

$[0, \frac{2}{3}]$
D

none of these

Solution

For the existence of the given equation, we must have
$- 1 \leq x \leq 1$ and $- 1 \leq 3 x - 1 \leq 1 \Rightarrow 0 \leq x \leq \frac{2}{3}$
Now,
$\begin{array}{l} \sin^{- 1} x = \cos^{- 1} x + \sin^{- 1} (3 x - 1) \\ \Rightarrow \sin^{- 1} x - \cos^{- 1} x = \sin^{- 1} (3 x - 1) \\ \Rightarrow 2 \sin^{- 1} x - \frac{π}{2} = \sin^{- 1} (3 x - 1) \\ \begin{array}{ll} \Rightarrow \sin (2 \sin^{- 1} x - \frac{π}{2}) = \sin (\sin^{- 1} (3 x - 1)) \\ \Rightarrow - \cos (2 \sin^{- 1} x) = 3 x - 1 \\ \Rightarrow - \{1 - 2 \sin^{2} (\sin^{- 1} x)\} = 3 x - 1 \\ \Rightarrow - (1 - 2 x^{2}) = 3 x - 1 \\ \Rightarrow 2 x^{2} - 3 x = 0 \Rightarrow x = 0, \frac{3}{2} \Rightarrow x = 0 [∵ 0 \leq x \leq 2 / 3] \end{array} \end{array}$

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