I simplified
$$\arcsin(x) -\arcsin(y) = \frac{\pi}{2}$$
as follows:
$$\arcsin(y) = \arcsin(x) -\frac{\pi}{2} \tag{1}$$
Taking $\sin$ on both sides:
$$\sin(\arcsin(y)) =-\sin\left(\frac{\pi}{2}-\arcsin(x)\right)$$
$$y=-\cos(\arcsin(x)) \tag{2}$$
The graph of $(1)$ (as shown in this Desmos graph) is the portion of the unit circle in the fourth quadrant. The graph of $(2)$ (Desmos) is a semicircle in the lower quadrants.
Since two graphs are different, both functions should be different. But I didn't alter the original function, I just simplified it. There must be some step (during simplification) where I changed the function unknowingly. But I'm not able to find the error I made. Please tell where I changed the function and how ?
Best Answer
Usually, arcsin has values in $[-\pi / 2, + \pi /2]$, which implies that your first equation is defined only for $0\leq x \leq 1$. If it is not clear, just remember that when taking sin on both sides , the equality obtained is not equivalent to the initial one, since $x \mapsto \sin (x)$ is only injective on some restriction of its domain.