Q70 / 160 · TS EAMCET 2004 · Mathematics

If $\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x$, then $x \in$ to

  • A
    $\{1,0\}$
  • B
    $\{-1,1\}$
  • Correct answer
    C
    $\left\{0, \frac{1}{2}\right\}$
  • D
    $\{2,0\}$

Solution

We have, $\begin{aligned} & \sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x \\ & \Rightarrow \sin ^{-1}(1-x)=\frac{\pi}{2}-\sin ^{-1} x-\sin ^{-1} x \\ & =\frac{\pi}{2}-2 \sin ^{-1} x \\ & {\left[\because \cos ^{-1} x+\sin ^{-1} x=\frac{\pi}{2}\right]} \\ & \Rightarrow \sin ^{-1}(1-x)=\sin ^{-1} 1-\sin ^{-1} 2 x \sqrt{1-x^2} \\ & \Rightarrow \sin ^{-1}(1-x)=\sin ^{-1}\left(1 \sqrt{1-4 x^2\left(1-x^2\right)}-0\right) \\ & \Rightarrow \quad(1-x)=\sqrt{1-4 x^4+4 x^2}=\sqrt{\left(1-2 x^2\right)^2} \\ & \Rightarrow \quad(1-x)=1-2 x^2 \\ & \Rightarrow \quad 2 x^2-x=0 \\ & \Rightarrow \quad x(2 x-1)=0 \\ & \Rightarrow \quad x=0, \frac{1}{2} \\ & \therefore \quad x \in\left\{0, \frac{1}{2}\right\} \\ & \end{aligned}$

Hence, the correct answer is option: 'C'.

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