Answer by steven gregory for Why is $\cos(x)^2$ written as $\cos^2(x)$?

There are several competing notations. These seem to be the standard interpretations. The goal seems to be to use the least number of parenthesis and still be understandable.

$\left . \begin{matrix} \cos(\cos(x)) \\ (\cos(x))^2 \\ \end{matrix} \right\} = \cos^2(x) = \cos(x)^2$

$\left . \begin{matrix} \dfrac{1}{cos(x)} \\ \arccos(x) \end{matrix} \right\} = \cos^{-1}(x)$

$\cos(x^2) = \cos(x)^2 = \cos x^2$

Please note that $\cos(x)^2$ is the most ambiguous of the group and I personally feel that it should be avoided as much as possible.

Generally, the context should make it clear which meaning is being used.