1. Inverse Issues

Is the inverse of f^(-1) (x) specifically sin^(-1) (x) [arcsinx] or does the inverse of f^(-1) (x) apply to every trigonometric inverse?

2. The inverse of f^(-1)(x) is f(x).

Maybe I don't understand the question.

3. Maybe I copied something wrong in my calculus notes?

4. Is the inverse of f^(-1) (x) ... like TK said.. the inverse of the inverse is the original function.

...specifically sin^(-1) (x) [arcsinx] ... yes, sin^(-1)(x) is also known as arcsin(x).

...does the inverse of f^(-1) (x) apply to every trigonometric inverse? yeah... like TK said. What are you talking about??

5. Strictly speaking, the trig functions don't have inverses because they are not "one to one". That is, $sin(0)= sin(\pi)= 0$ so what should $sin^{-1}(0)$ be?. What we can do is restrict the functions: if we restrict x to be between $-\pi$ and $\pi$ then It is one to one and so has an inverse (and $sin^{-1}(0)= 0$).

The notation, $f^{-1}(x)$, can be used for the inverse of any function that has an inverse. "arcsin(x)" is a different notation that means exactly the same as " $sin^{-1}(x)$".