Why is $\cos(x)^2$ written as $\cos^2(x)$?

Because we can safely drop the brackets without losing ambiguity, which means less effort when writing it out by hand. $$ \cos^2x = \cos^2(x) = \cos(x)^2 \neq \cos x^2 $$

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.

This is because composition of functions are very rare when you are talking about trigonometric functions.

For any other $f: \mathbb{D} \to \mathbb{R}$, it may make sense to calculate $f(f(x))$, however for $\sin(x)$ or $\cos(x)$, composition like $\cos(\cos(x))$ is not a frequent use. That's why a misunderstanding in $\cos^2(x)$ is not so much in concern.

On the other hand, when it is about $\arcsin(x)$ and $\csc(x)$, there are conflicts about the use of $\sin^{-1}(x)$.