Question

Why do the functions f(x) = sin−1(x) and g(x) = cos−1(x) have different ranges?

Answer 1

We Know that
For a function to have an inverse function, it must be one-to-one—that is, it must pass the Horizontal Line Test.

1. On the interval [–pi/2, pi/2], the function y = sin x is increasing
2. On the interval [–pi/2, pi/2], y = sin x takes on its full range of values, [–1, 1]
3. On the interval [–pi/2, pi/2], y = sin x is one-to-one
sin x has an inverse function on this interval [–pi/2, pi/2]

On the restricted domain [–pi/2, pi/2]  y = sin x has a unique inverse function called the inverse sine function. f(x) = sin−1(x)
the range of y=sin x  in the domain [–pi/2, pi/2]  is [-1,1] 
the range of y=sin-1  x  in the domain [-1,1]  is [–pi/2, pi/2]  

1. On the interval [0, pi], the function y = cos x is decreasing
2. On the interval [0, pi], y = cos x takes on its full range of values, [–1, 1]
3. On the interval [0, pi], y = cos x is one-to-one
cos x has an inverse function on this interval [0, pi]

On the restricted domain [0, pi]  y = cos x has a unique inverse function called the inverse sine function. f(x) = cos−1(x)
the range of y=cos x  in the domain [0, pi]  is [-1,1] 
the range of y=cos-1  x  in the domain [-1,1]  is [0, pi] 

the answer is

the values ​​of the range are different because the domain in which the inverse function exists are different