How many solutions does $2 (\sin^{-1}x)^2 -5\sin^{-1}x+2=0$ have? 
Number of solutions of the equation
$2 (\sin^{-1}x)^2 -5\sin^{-1}x+2=0 $

Let
$t=\sin^{-1}x $
$2 t^2 -5t+2=0$
$(t-2)(2t-1)=0 $
$t=2$    or $t=1/2$
Then
$\sin^{-1}x=2\quad$  or$\quad   \sin^{-1}x=1/2$
$x=\sin 2\quad$  or$\quad  x=\sin(1/2)$
Did I do something wrong here? I found a solution here =>  https://www.toppr.com/ask/question/number-of-solutions-of-the-equation-2sin1x25sin1x20/
The solution given on this site says only one solution exists. But I can't understand why $x=\sin 2$   is rejecting. isn't $t$ is an angle and $x$ is a value? Please help.
 A: You have to reject $x=\sin 2$ as $\arcsin x$ is defined on $[-1,1]$ and takes values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ by definition. As $2$ doesn’t belong to the last interval, this solution has to be rejected.
See inverse trigonometric functions.
A: The (real-valued) arcsine returns a number in $[-\pi/2,\pi/2]$. In particular $\arcsin\sin2$ is not $2$, but $\pi-2$, so it doesn't work.
A: I disagree with the other responses as well as the official answer, because I consider the normal range of the ArcSin function as irrelevant.
Consider the equation $t^2 = 2.$  The fact that the square root function has a normal range of non-negative values only is irrelevant.  The equation $t^2 = 2$ still has two solutions, not one.
Just because the ArcSin function normally has a restricted range does not render the equation ArcSin$(x) = 2$ as gibberish.  Here, $x$ can be reasonably interpreted as representing the $\sin(2)$.
Ultimately, it comes down to interpretation.  Do you interpret the equation ArcSin$(x) = 2$ to mean that $2$ is the result of the ArcSin function at some value $x$ (which would imply that this equation has no solution), or do you interpret the equation as find $x$ such that $\sin(2) = x.$
Note that the $\sin^{(-1)}$ syntax is being used instead of ArcSin.  It is not that unusual, when faced with a non-injective function $f(x)=y$ to interpret $f^{(-1)}(y)$ to represent all of the values of $x$ such that $f(x) = y.$
Very similarly, do you interpret the equation $t^2 = 2$ as find $t$ such that $t^2 = 2$, or do you interpret it as find $t$ such that $t = \sqrt{2}.$
