By doing some right triangle gymnastics, we can derive things like
$\cos(\arctan x) = \frac{1}{\sqrt{1+x^2}}$, for $x>0$
$\cos(\arcsin x) = \sqrt{1-x^2}$
$\tan(\arcsin x) = \frac{x}{\sqrt{1-x^2}}$
What about $\arctan\cos(x)$, $\arcsin(\tan x)$, etc? I understand that in this case $x$ is treated as an angle, not a ratio of side lengths and that it is impossible to construct the same kind of right triangle relations for these formulas. However, is there a particularly compelling non-geometric reason why the reverse application of these functions is intractable?