Question

The number of solution of the equation $si{n}^{-1}x+si{n}^{-1}(1-x)=co{s}^{-1}x$ is

• A. $1$
• B. $0$
• C. $2$
• D. $noneofthese$

Answer 1

Answer: (C)

$2$

$si{n}^{-1}\left(x\right)+si{n}^{-1}(1-x)$
$=si{n}^{-1}(x\sqrt{2x-x^2}+(1-x\left)\sqrt{1-x^2}\right)$
$=si{n}^{-1}\left(\sqrt{1-x^2}\right)$
Hence
$x\sqrt{2x-x^2}+(1-x)\sqrt{1-x^2}=\sqrt{1-x^2}$
$x\sqrt{2x-x^2}+\sqrt{1-x^2}-x\sqrt{1-x^2}=\sqrt{1-x^2}$
$x\sqrt{2x-x^2}=x\sqrt{1-x^2}$
Squaring on both the sides, we get
$x^2(2x-x^2)=x^2(1-x^2)$
$x^2(2x-x^2-1+x^2)=0$
$x^2(2x-1)=0$
Therefore
$x=0$ and $x=\frac{1}{2}$
Hence there are 2, solutions, to the above equation.