Answer:
It is given that : [tex]Sin^{-1}(Sin (x) =x[/tex]
As we have to find the Domain of [tex]Sin^{-1}(Sin (x)[/tex].
-1 ≤ sin x ≤1
[tex]Sin^{-1}(-1)\leq Sin^{-1}(Sin x)\leq Sin^{-1}(1)\\\frac{-\pi}{2}\leq x\leq \frac{\pi }{2}[/tex]
So it is defined for x lying between [tex](\frac{-\pi}{2}, \frac{\pi }{2})[/tex] .
As meaning of Domain is those values of x for which any function f(x) is defined.
The expression is valid for x within the interval [tex](-\frac{\pi}{2}, \frac{\pi}{2} )[/tex]
For the cancellation property to exist, it means that the value of the domain x must lie between - 1 and 1 that is:
[tex]-1\leq sinx \leq1[/tex]
Given the property;
[tex]sin^{-1}(sinx)=x[/tex]
If sinx = 1
[tex]x = sin^{-1}1\\x = 90^0 = \frac{\pi}{2}[/tex]
Similarly, if sinx = -1;
[tex]x = sin^{-1}-1\\x = -90^0 = -\frac{\pi}{2}[/tex]
This shows that the expression is valid for x within the interval [tex](-\frac{\pi}{2}, \frac{\pi}{2} )[/tex]
Learn more here: https://brainly.com/question/24493211
