TITLE: Why $\arctan x$ not equal to $\arcsin(x)/\arccos(x)$?
QUESTION [5 upvotes]: Why $\arctan x$ not equal to $\frac{\arcsin(x)}{\arccos(x)}$? Is there a counter example that I can use to show that they are not equal? Thank!

REPLY [3 votes]: It's not terribly difficult to show if $h(x)=\frac{f(x)}{g(x)}$, then, in general,
$$h^{-1}(x) \neq \frac{f^{-1}(x)}{g^{-1}(x)}$$
Or, more generally, if $g(x)=f_1\circ ... \circ f_n (x)$,
$$g^{-1}(x) \neq f_1^{-1}\circ ... \circ f_n^{-1}(x).$$