First slide

Properties of ITF

Question

$The x satisfying$ $\sin^{- 1} x + \sin^{- 1} (1 - x) = \cos^{- 1} x$ are

Moderate

A

$1, 0$
B

$1, - 1$
C

$0, \frac{1}{2}$
D

None of these

Solution

$\begin{matrix} \sin^{- 1} (1 - x) = (\frac{π}{2} - \sin^{- 1} x) - \sin^{- 1} x (∵ \cos^{- 1} x = \frac{π}{2} - \sin^{- 1} x) \\ \sin^{- 1} (1 - x) = \frac{π}{2} - 2 \sin^{- 1} x \\ \Rightarrow 1 - x = \sin (\frac{π}{2} - 2 \sin^{- 1} x) = \cos (2 \sin^{- 1} x) \\ = \cos 2 θ, where \sin^{- 1} x = θ \\ \Rightarrow 1 - x = 1 - 2 \sin^{2} θ = 1 - 2 x^{2} or x (1 - 2 x) = 0 or x = 0, \frac{1}{2} \end{matrix}$

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