First slide

Properties of ITF

Question

The number of solutions of $\sin^{- 1} x + \sin^{- 1} (1 - x) = \cos^{- 1} x$ is

Moderate

A

1
B

0
C

2
D

4

Solution

$\begin{array}{l} \sin^{- 1} x + \sin^{- 1} (1 - x) = \cos^{- 1} x \\ \Rightarrow \sin^{- 1} (1 - x) = \cos^{- 1} x - \sin^{- 1} x = \frac{π}{2} - 2 \sin^{- 1} x \end{array}$
$\begin{array}{l} \Rightarrow 1 - x = \cos (2 \sin^{- 1} x) = 1 - 2 {(\sin \sin^{- 1} x)}^{2} = 1 - 2 x^{2} \\ \Rightarrow 2 x^{2} - x = 0 \Rightarrow x = 0 or \frac{1}{2} \end{array}$

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